hcju/proofwikips huggingface dataset:
Definition:Differential Operator
Let $A$ be a mapping from a function space $\FF_1$ to another function space $\FF_2$. Let $f \in \FF_2$ be a real function such that $f$ is the image of $u \in \FF_1$ that is:　$f = A \sqbrk u$ A '''differential operator''' is represented as a linear combination, finitely generated by $u$ and its derivatives containing higher degree such as :$\displaystyle \map P {x, D} = \sum _{\size \alpha \mathop \le m} \map {a_\alpha} x D^\alpha$ where: :$\alpha = \set {\alpha_1, \alpha_2, \dotsc \alpha_n}$ is a set of non-negative integers forming a multi-index :$\size \alpha = \alpha_1 + \alpha_2 + \dotsb + \alpha_n$ is the length of $\alpha$ :the $\map {a_\alpha} x$ are real functions on a open domain in a real cartesian space of $n$ dimensions :$D^\alpha = D_1^{\alpha_1} D_2^{\alpha_2} \dotsm D_n^{\alpha_n}$. {{Proofread|This definition copied from Wikipedia and made more or less coherent. Please correct as necessary.}}
Simple Events are Mutually Exclusive
Let $\EE$ be an experiment. Let $e_1$ and $e_2$ be distinct simple events in $\EE$. Then $e_1$ and $e_2$ are mutually exclusive.
Non-Trivial Event is Union of Simple Events
Let $\EE$ be an experiment. Let $e$ be an event in $\EE$ such that $e \ne \O$. That is, such that $e$ is non-trivial. Then $e$ can be expressed as the union of a set of simple events in $\EE$.
Sample Space is Union of All Distinct Simple Events
Let $\EE$ be an experiment. Let $\Omega$ denote the sample space of $\EE$. Then $\Omega$ is the union of the set of simple events in $\EE$.
Equivalence of Definitions of Probability Measure
Let $\EE$ be an experiment. {{TFAE|def = Probability Measure}}
Product Space is Product in Category of Topological Spaces
Let $\mathbf{Top}$ be the category of topological spaces. Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $\struct{X, \tau}$ be the product space of $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$. Then $\struct{X, \tau}$ is the product of $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ in $\mathbf{Top}$.
Strictly Positive Rational Numbers are Closed under Addition
:$\forall a, b \in \Q_{>0}: a + b \in \Q_{>0}$
Strictly Positive Rational Numbers are Closed under Multiplication
:$\forall a, b \in \Q_{>0}: a b \in \Q_{>0}$
Set of Rationals Greater than Root 2 has no Smallest Element
Let $B$ be the set of all positive rational numbers $p$ such that $p^2 > 2$. Then $B$ has no smallest element.
Rational Number Not in Cut is Greater than Element of Cut
Let $\alpha$ be a cut. Let $p \in \alpha$. Let $q \in \Q$ such that $q \notin \alpha$. Then $q > p$.
Natural Basis of Tychonoff Topology
Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $X$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$: :$\displaystyle X := \prod_{i \mathop \in I} X_i$ Then the natural basis on $X$ is the set $\BB$ of cartesian products of the form $\displaystyle \prod_{i \mathop \in I} U_i$ where: :for all $i \in I : U_i \in \tau_i$ :for all but finitely many indices $i : U_i = X_i$
Natural Basis of Tychonoff Topology/Finite Product
Let $n \in \N$. For all $k \in \set {1, \ldots, n}$, let $\struct {X_k, \tau_k}$ be topological spaces. Let $\displaystyle X = \prod_{k \mathop = 1}^n X_k$ be the cartesian product of $X_1, \ldots, X_n$. Then the natural basis on $X$ is: :$\BB = \set {\displaystyle \prod_{k \mathop = 1}^n U_k : \forall k : U_k \in \tau_k}$
Box Topology may not be Coarsest Topology such that Projections are Continuous
Let $\family {\struct{X_i, \tau_i}}_{i \mathop \in I}$ be an $I$-indexed family of topological spaces. Let $X$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$, that is: :$\displaystyle X := \prod_{i \mathop \in I} X_i$ Let $\tau$ be the box topology on $X$. For each $i \in I$, let $\pr_i: X \to X_i$ denote the $i$th projection on $X$: :$\forall \family {x_j}_{j \mathop \in I} \in X: \map {\pr_i} {\family {x_j}_{j \mathop \in I} } = x_i$ Then $\tau$ may not be the coarsest topology on $X$ for which the projections $\family{\pr_i} _{i \mathop \in I}$ are continuous.
Box Topology may not form Categorical Product in the Category of Topological Spaces
Let $\family {\struct{X_i, \tau_i}}_{i \mathop \in I}$ be an $I$-indexed family of topological spaces. Let $X$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$, that is: :$\displaystyle X := \prod_{i \mathop \in I} X_i$ Let $\tau$ be the box topology on $X$. Then $\tau$ may not be the categorical product in the category of topological spaces.
Ordering on Cuts satisfies Trichotomy Law
Let $\alpha$ and $\beta$ be cuts. Then exactly one of the following applies: {{begin-eqn}} {{eqn | n = 1       | l = \alpha       | o = <       | r = \beta }} {{eqn | n = 2       | l = \alpha       | r = \beta }} {{eqn | n = 3       | l = \alpha       | o = >       | r = \beta }} {{end-eqn}} where $<$ and so $>$ denote the strict ordering of cuts: :$\alpha < \beta \iff \exists p \in \Q: p \in \alpha, p \notin \beta$ Hence the ordering of cuts $\le$ is a total ordering.
Ordering on Cuts is Transitive
Let $\alpha$, $\beta$ and $\gamma$ be cuts. Let: {{begin-eqn}} {{eqn | n = 1       | l = \alpha       | o = <       | r = \beta }} {{eqn | n = 2       | l = \beta       | o = <       | r = \gamma }} {{end-eqn}} where $<$ denotes the strict ordering of cuts: :$\alpha < \beta \iff \exists p \in \Q: p \in \alpha, p \notin \beta$ Then: :$\alpha < \gamma$ Hence the ordering of cuts $\le$ is a transitive relation.
Ordering on Cuts is Total
Let $\CC$ denote the set of cuts. Let $<$ denote the strict ordering on cuts defined as: :$\forall \alpha, \beta \in \CC: \alpha < \beta \iff \exists p \in \Q: p \in \alpha, p \notin \beta$ Then $<$ is a (strict) total ordering on $\CC$.
Sum of Cuts is Cut
Let $\alpha$ and $\beta$ be cuts. Let $\gamma$ be the set of all rational numbers $r$ such that: :$\exists p \in \alpha, q \in \beta: r = p + q$ Then $\gamma$ is also a cut. Thus the operation of addition on the set of cuts is closed.
Addition of Cuts is Commutative
Let $\alpha$ and $\beta$ be cuts. Let the operation of $\alpha + \beta$ be the sum of $\alpha$ and $\beta$. Then: :$\alpha + \beta = \beta + \alpha$
Addition of Cuts is Associative
Let $\alpha$, $\beta$ and $\gamma$ be cuts. Let the operation of $\alpha + \beta$ be the sum of $\alpha$ and $\beta$. Then: :$\paren {\alpha + \beta} + \gamma = \alpha + \paren {\beta + \gamma}$
Identity Element for Addition of Cuts
Let $\alpha$ be a cut. Let $0^*$ be the rational cut associated with the (rational) number $0$: :$0^* = \set {r \in \Q: r < 0}$ Then: :$\alpha + 0^* = \alpha$ where $+$ denotes the operation of addition of cuts.
Existence of Upper and Lower Numbers of Cut whose Difference equal Given Rational
Let $\alpha$ be a cut. Let $r \in \Q_{>0}$ be a (strictly) positive rational number. Then there exist rational numbers $p$ and $q$ such that: :$p \in \alpha, q \notin \alpha$ :$q - p = r$ such that $q$ is not the smallest upper number of $\alpha$.
Existence of Unique Inverse Element for Addition of Cuts
Let $\alpha$ be a cut. Let $0^*$ be the rational cut associated with the (rational) number $0$: :$0^* = \set {r \in \Q: r < 0}$ Then there exists a unique cut $\beta$ such that: :$\alpha + \beta = 0^*$ where $+$ denotes the operation of addition of cuts.
Ordering on Cuts is Compatible with Addition of Cuts
Let $\alpha$, $\beta$ and $\gamma$ be cuts. Let the operation of $\alpha + \beta$ be the sum of $\alpha$ and $\beta$. Let $\beta < \gamma$ denotes the strict ordering on cuts defined as: :$\beta < \gamma \iff \exists p \in \Q: p \in \beta, p \notin \gamma$ Then: :$\beta < \gamma \implies \alpha + \beta < \alpha + \gamma$
Ordering on Cuts is Compatible with Addition of Cuts/Corollary
Let $0^*$ denote the rational cut associated with the (rational) number $0$. If: :$\alpha > 0^*$ and $\gamma > 0^*$ then: :$\alpha + \gamma > 0^*$
Existence of Unique Difference between Cuts
Let $\alpha$ and $\beta$ be cuts. Then there exists exactly one cut $\gamma$ such that: :$\alpha + \gamma = \beta$
Set of Cuts under Addition forms Abelian Group
Let $\CC$ denote the set of cuts. Let $\struct {\CC, +}$ denote the algebraic structure formed from $\CC$ and the operation $+$ of addition of cuts. Then $\struct {\CC, +}$ forms an abelian group.
Product of Positive Cuts is Positive Cut
Let $0^*$ denote the rational cut associated with the (rational) number $0$. Let $\alpha$ and $\beta$ be cuts such that $\alpha \ge 0^*$ and $\beta \ge 0^*$, where $\ge$ denotes the ordering on cuts. Let $\gamma$ be the set of all rational numbers $r$ such that either: :$r < 0$ or: :$\exists p \in \alpha, q \in \beta: r = p q$ where $p \ge 0$ and $q \ge 0$. Then $\gamma$ is also a cut. Thus the operation of multiplication on the set of positive cuts is closed.
Product of Cuts is Cut
Let $\alpha$ and $\beta$ be cuts. Let $\alpha \beta$ denote the product of cuts. Then $\alpha \beta$ is also a cut. Thus the operation of multiplication on the set of cuts is closed.
Factorisation of z^n+a
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $a \in \C$ be a complex number. Then: :$z^n + a = \displaystyle \prod_{k \mathop = 0}^{n - 1} \paren {z - \alpha_k b}$ where: :$\alpha_k$ are the complex $n$th roots of negative unity :$b$ is any complex number such that $b^n = a$.
Multiplication of Cuts is Commutative
Let $\alpha$ and $\beta$ be cuts. Let the $\alpha \beta$ be the product of $\alpha$ and $\beta$. Then: :$\alpha \beta = \beta \alpha$
Multiplication of Cuts is Associative
Let $\alpha$, $\beta$ and $\gamma$ be cuts. Let $\alpha \beta$ denote the product of $\alpha$ and $\beta$. Then: :$\paren {\alpha \beta} \gamma = \alpha \paren {\beta \gamma}$
Natural Basis of Tychonoff Topology/Lemma 1
:$\SS \subseteq \BB$
Natural Basis of Tychonoff Topology/Lemma 2
:$\forall B_1, B_2 \in \BB : B_1 \cap B_2 \in \BB$
Natural Basis of Tychonoff Topology/Lemma 3
:$\displaystyle \forall B \in \BB : B = \bigcap_{j \mathop \in J} \pr_j^{-1} \sqbrk {U_j}$ where: :$\displaystyle B = \prod_{i \mathop \in I} U_i$ :$J = \set{j \in I : U_i \ne X_i}$ is finite.
Multiplication of Cuts Distributes over Addition
Let $\alpha$, $\beta$ and $\gamma$ be cuts. Let: :$\alpha + \beta$ denote the sum of $\alpha$ and $\beta$. :$\alpha \beta$ denote the product of $\alpha$ and $\beta$. Then: :$\alpha \paren {\beta + \gamma} = \alpha \beta + \alpha \gamma$
Product of Cut with Zero Cut equals Zero Cut
Let $\alpha$ be a cut. Let $0^*$ denote the rational cut associated with the (rational) number $0$. Then: :$\alpha 0^* = 0^*$ where $\alpha 0^*$ denote the product of $\alpha$ and $0^*$.
Product of Cuts is Zero Cut iff Either Factor equals Zero Cut
Let $\alpha$ and $\beta$ be cuts. Let $0^*$ denote the rational cut associated with the (rational) number $0$. Then: :$\alpha \beta = 0^*$ : :$\alpha = 0^*$ or $\beta = 0^*$ where $\alpha \beta$ denotes the product of $\alpha$ and $\beta$.
Cut Associated with 1 is Identity for Multiplication of Cuts
Let $\alpha$ be a cut. Let $1^*$ denote the rational cut (rational) number $1$. Then: :$\alpha 1^* = \alpha$ where $\alpha 1^*$ denote the product of $\alpha$ and $1^*$.
Multiplication of Positive Cuts preserves Ordering
Let $0^*$ denote the rational cut associated with the (rational) number $0$. Let $\alpha$, $\beta$ and $\gamma$ be cuts such that: :$0^* < \alpha < \beta$ :$0^* < \gamma$ where $<$ denotes the strict ordering on cuts. Then :$\alpha \gamma < \beta \gamma$ where $\alpha \gamma$ denotes the product of $\alpha$ and $\gamma$.
Existence of Unique Inverse Element for Multiplication of Cuts
Let $0^*$ be the rational cut associated with the (rational) number $0$: :$0^* = \set {r \in \Q: r < 0}$ Let $\alpha$ be a cut such that $\alpha \ne 0^*$. Then for every cut $\beta$, there exists a unique cut $\gamma$ such that: :$\alpha \gamma = \beta$ where $\alpha \gamma$ denotes the operation of product of $\alpha$ and $\gamma$. In this context, $\gamma$ can be expressed as $\beta / \alpha$.
Sum of Rational Cuts is Rational Cut
Let $p \in\ Q$ and $q \in \Q$ be rational numbers. Let $p^*$ and $q^*$ denote the rational cuts associated with $p$ and $q$. Then: :$p^* + q^* = \paren {p + q}^*$ Thus the operation of addition on the set of rational cuts is closed.
Ordering of Rational Cuts preserves Ordering of Associated Rational Numbers
Let $p \in\ Q$ and $q \in \Q$ be rational numbers. Let $p^*$ and $q^*$ denote the rational cuts associated with $p$ and $q$. Then: :$p^* < q^* \iff p < q$ where $p^* < q^*$ denotes the strict ordering on cuts defined as: :$\beta < \gamma \iff \exists p \in \Q: p \in \beta, p \notin \gamma$
Exists Rational Cut Between two Cuts
Let $\alpha$ and $\beta$ be cuts. Let $\alpha < \beta$, where $<$ denotes the strict ordering on cuts. Then there exists a rational cut $r^*$ associated with the rational number $r$ such that: :$\alpha < r^* < \beta$
Condition for Rational Cut to be Less than Given Cut
Let $\alpha$ be a cut. Let $p^*$ be the rational cut associated with a rational number $p$. Then: :$p \in \alpha$ : :$p^* < \alpha$ where $<$ denotes the strict ordering on cuts.
Continuous Mapping to Topological Product/General Result
Let $X$ be a topological space. Let $\family {Y_i}_{i \mathop \in I}$ be an indexed family of topological spaces for some indexing set $I$. Let $\displaystyle Y = \prod_{i \mathop \in I} Y_i$ be the product space of $\family {Y_i}_{i \mathop \in I}$.  For each $i \in I$, let $\pr_i: X \to X_i$ denote the $i$th projection on $X$: :$\forall \family {x_j}_{j \mathop \in I} \in X: \map {\pr_i} {\family {x_j}_{j \mathop \in I} } = x_i$ Let $f$ be a mapping from $X$ to $Y$.  Then $f$ is continuous  $\pr_i \circ f$ is continuous for all $i \in I$.
Set of Cuts forms Ordered Field
Let $\CC$ denote the set of cuts. Let $\struct {\CC, +, \times, \le}$ denote the ordered structure formed from $\CC$ and: :the operation $+$ of addition of cuts :the operation $\times$ of multiplication of cuts :the ordering $\le$ of cuts. Then $\struct {\CC, +, \times, \le}$ is an ordered field.
Box Topology contains Tychonoff Topology
Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $X$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$: :$\displaystyle X := \prod_{i \mathop \in I} X_i$ Let $\tau$ be the Tychonoff topology on $X$. Let $\tau'$ be the box topology on $X$. Then: :$\tau \subseteq \tau'$
Ordered Field of Rational Cuts is Isomorphic to Rational Numbers
Let $\struct {\RR, +, \times, \le}$ denote the ordered field of rational cuts. Let $\struct {\Q, +, \times, \le}$ denote the field of rational numbers. Then $\struct {\RR, +, \times, \le}$ and $\struct {\Q, +, \times, \le}$ are isomorphic.
Supremum of Subset of Real Numbers May or May Not be in Subset
Let $S \subset \R$ be a proper subset of the set $\R$ of real numbers. Let $S$ admit a supremum $M$. Then $M$ may or may not be an element of $S$.
Uniqueness of Positive Root of Positive Real Number/Positive Exponent
Let $x \in \R$ be a real number such that $x > 0$. Let $n \in \Z$ be an integer such that $n > 0$. Then there is at most one $y \in \R: y \ge 0$ such that $y^n = x$.
Basis Condition for Coarser Topology
Let $S$ be a set. Let $\BB_1$ and $\BB_2$ be two bases on $S$. Let $\tau_1$ and $\tau_2$ be the topologies generated by $\BB_1$ and $\BB_2$ respectively. If $\BB_1$ and $\BB_2$ satisfy: :$\forall U \in \BB_1 : \exists \AA \subseteq \BB_2: U = \bigcup \AA$ then $\tau_1$ is coarser than $\tau_2$.
Basis Condition for Coarser Topology/Corollary 2
If $\BB_1 \subseteq \BB_2$ then $\tau_1$ is coarser than $\tau_2$.
Continuous Mapping on Finer Domain and Coarser Codomain Topologies is Continuous
Let $\struct {X, \tau_1}$ and $\struct {Y, \tau_2}$ be topological spaces. Let $f: \struct {X, \tau_1} \to \struct {Y, \tau_2}$ be a continuous mapping. Let $\tau'_1$ be a finer topology on $X$ than $\tau_1$, that is, $\tau_1 \subseteq \tau'_1$. Let $\tau'_2$ be a coarser topology on $Y$ than $\tau_2$, that is, $\tau'_2 \subseteq \tau_2$. Then: :$f: \struct {X, \tau'_1} \to \struct {Y, \tau'_2}$ is a continuous mapping.
Projection from Box Topology is Continuous
Let $\family {\struct{X_i, \tau_i}}_{i \mathop \in I}$ be an $I$-indexed family of topological spaces. Let $X$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$, that is: :$\displaystyle X := \prod_{i \mathop \in I} X_i$ Let $\tau$ be the box topology on $X$. For each $i \in I$, let $\pr_i: X \to X_i$ denote the $i$th projection on $X$: :$\forall \family {x_j}_{j \mathop \in I} \in X: \map {\pr_i} {\family {x_j}_{j \mathop \in I} } = x_i$ Then $\pr_i: \struct{X,\tau} \to \struct{X_i,\tau_i}$ is continuous for all $i \in I$.
Domain Topology Contains Initial Topology iff Mappings are Continuous
Let $\struct{Y, \tau}$ be a topological space. Let $\family {\struct{X_i, \tau_i}}_{i \mathop \in I}$ be a family of topological spaces. Let $\family {f_i}_{i \mathop \in I}$ be a family of mappings $f_i : Y \to X_i$. Let $\tau'$ be the initial topology on $Y$ with respect to $\family {f_i}_{i \mathop \in I}$. Then: :$\tau' \subseteq \tau$  $\forall i \in I : f_i: \struct{Y, \tau} \to \struct{X_i, \tau_i}$ is $\tuple{\tau, \tau_i}$-continuous.
Final Topology Contains Codomain Topology iff Mappings are Continuous
Let $\struct{Y, \tau}$ be a topological space. Let $\family {\struct{X_i, \tau_i}}_{i \mathop \in I}$ be a family of topological spaces. Let $\family {f_i}_{i \mathop \in I}$ be a family of mappings $f_i : X_i \to Y$. Let $\tau'$ be the final topology on $Y$ with respect to $\family {f_i}_{i \mathop \in I}$. Then: :$\tau \subseteq \tau'$  $\forall i \in I : f_i: \struct{X_i, \tau_i} \to \struct{Y, \tau}$ is continuous.
Equivalence of Definitions of Final Topology
Let $X$ be a set. Let $I$ be an indexing set. Let $\family {\struct {Y_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces indexed by $I$. Let $\family {f_i: Y_i \to X}_{i \mathop \in I}$ be an indexed family of mappings indexed by $I$. {{TFAE|def = Final Topology}}
Condition for Trivial Relation to be Mapping
Let $S$ and $T$ be sets. Let $\RR = S \times T$ be the trivial relation in $S$ to $T$. Then $\RR$ is a mapping  either: :$(2): \card S = 0$ or: :$(1): \card T = 1$ where $\card {\, \cdot \,}$ denotes cardinality.
Relation to Empty Set is Mapping iff Domain is Empty
Let $S$ be a set Let $S \times \O$ denote the cartesian product of $S$ with the empty set $\O$. Let $\RR \subseteq S \times \O$ be a relation in $S$ to $\O$. Then $\RR$ is a mapping  $S = \O$.
Equivalence of Definitions of Initial Topology/Definition 1 Implies Definition 2
Let: :$\SS = \set{\map {f_i^{-1}} U: i \in I, U \in \tau_i} \subseteq \map \PP X$ where $\map {f_i^{-1}} U$ denotes the preimage of $U$ under $f_i$. Let $\tau$ be the topology on $X$ generated by the subbase $\SS$.
Equivalence of Definitions of Initial Topology/Definition 2 Implies Definition 1
Let $\tau$ be the coarsest topology on $X$ such that each $f_i: X \to Y_i$ is $\tuple{\tau, \tau_i}$-continuous. Let: :$\SS = \set {\map {f_i^{-1} } U: i \in I, U \in \tau_i} \subseteq \map \PP X$ where $\map {f_i^{-1} } U$ denotes the preimage of $U$ under $f_i$.
Even Natural Numbers are Infinite
The set of even natural numbers is infinite.
Set of Points on Line Segment is Infinite
The set of points on a line segment is infinite.
Finite Sets are Comparable
Let $S$ and $T$ be finite sets. Then $S$ and $T$ are comparable by size.
Equivalence of Definitions of Final Topology/Definition 1 Implies Definition 2
Let: :$\tau = \set{U \subseteq X: \forall i \in I: \map {f_i^{-1}} U \in \tau_i} \subseteq \powerset X$
Equivalence of Definitions of Final Topology/Definition 2 Implies Definition 1
Let $\tau$ be the finest topology on $X$ such that each $f_i: Y_i \to X$ is $\tuple{\tau_i, \tau}$-continuous.
Strictly Positive Integers have same Cardinality as Natural Numbers
Let $\Z_{>0} := \set {1, 2, 3, \ldots}$ denote the set of strictly positive integers. Let $\N := \set {0, 1, 2, \ldots}$ denote the set of natural numbers. Then $\Z_{>0}$ has the same cardinality as $\N$.
Power Set of Natural Numbers is Cardinality of Continuum
Let $\N$ denote the set of natural numbers. Let $\powerset \N$ denote the power set of $\N$. Let $\card {\powerset \N}$ denote the cardinality of $\powerset \N$. Then $\card {\powerset \N}$ is the cardinality of the continuum.
Continuum Hypothesis is Independent of ZFC
The Continuum Hypothesis can be neither proved nor disproved from the axioms of either Zermelo-Fraenkel set theory (ZF) or ZFC.
Axiom of Choice is Independent of ZF
The Axiom of Choice can be neither proved nor disproved from the axioms of Zermelo-Fraenkel set theory.
Empty Set can be Derived from Comprehension Principle
The '''empty set''' can be formed by application of the comprehension principle. Hence the '''empty set''' can be derived as a valid object in Frege set theory.
Power Set can be Derived using Comprehension Principle
Let $a$ be a set. By application of the comprehension principle, the power set $\powerset a$ can be formed. Hence the '''power set''' $\powerset a$ can be derived as a valid object in Frege set theory.
Set Union can be Derived using Comprehension Principle
Let $a$ be a set of sets. By application of the comprehension principle, the union $\bigcup a$ can be formed. Hence the '''union''' $\bigcup a$ can be derived as a valid object in Frege set theory.
Set of Natural Numbers can be Derived using Comprehension Principle
Let $\N$ denote the set of natural numbers. By application of the comprehension principle, $\N$ can be derived as a valid object in Frege set theory.
Exists Subset which is not Element
Let $S$ be a set. Then there exists at least one subset of $S$ which is not an element of $S$.
Box Topology may not be Coarsest Topology such that Projections are Continuous/Lemma
Let $\struct{X, \tau}$ be a topological space. Let $U \in \tau$ such that $U \neq \O$ and $U \neq X$. Let: :$Y = \displaystyle \prod_{n \mathop \in \N } X = X \times X \times X \times \ldots$ be the countable Cartesian product of $\family {X}_{n \in \N}$. Let $\tau_T$ be the Tychonoff topology on $Y$. Let $\tau_b$ be the box topology on $Y$. Let: :$V = \displaystyle \prod_{n \mathop \in \N } U = U \times U \times U \times \ldots$ be the countable Cartesian product of $\family {U}_{n \in \N}$. Then: :$V$ is an element of the box topology $\tau_b$ :$V$ is not an element of the Tychonoff topology $\tau_T$
P-adic Expansion Representative of P-adic Number is Unique
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. Let $\mathbf a$ be an equivalence class in $\Q_p$. Let $\displaystyle \sum_{i \mathop = m}^\infty d_i p^i$ and $\displaystyle \sum_{i \mathop = k}^\infty e_i p^i$ be $p$-adic expansions that represent $\mathbf a$. Then: :$(1) \quad m = k$ :$(2) \quad \forall i \ge m : d_i = e_i$ That is, the $p$-adic expansions $\displaystyle \sum_{i \mathop = m}^\infty d_i p^i$ and $\displaystyle \sum_{i \mathop = k}^\infty e_i p^i$ are identical.
Class is Subclass of Universal Class
Let $V$ denote the universal class. Let $A$ be a class. Then $A$ is a subclass of $V$.
Not Every Class is a Set
Let $A$ be a class. Then it is not necessarily the case that $A$ is also a set.
Class has Subclass which is not Element
Let $A$ be a class. Then $A$ has at least one subclass $B$ which is not an element of $A$.
Basic Universe is Supercomplete
Let $V$ be a basic universe. Then $V$ is supercomplete.
Basic Universe is not Set
Let $V$ be a basic universe. Then $V$ is not a set.
Empty Class is Subclass of All Classes
The empty class is a subclass of all classes.
Empty Class is Supercomplete
The empty class is supercomplete.
Basic Universe is not Empty
Let $V$ be a basic universe Then $V$ is not the empty class.
Singleton Class can be Formed from Set
Let $V$ be a basic universe. Let $a \in V$ be a set. Then the singleton class $\set a$ can be formed, which is a subclass of $V$.
Singleton Class of Empty Set is Supercomplete
Let $\O$ denote the empty set. Then the singleton $\set \O$ is supercomplete.