hcju/proofwikips huggingface dataset:
Category of Relations is Category
Let $\mathbf{Rel}$ be the category of relations. Then $\mathbf{Rel}$ is a metacategory.
Identity Functor is Functor
Let $\mathbf C$ be a metacategory. Let $\operatorname{id}_{\mathbf C}: \mathbf C \to \mathbf C$ be the identity functor on $\mathbf C$. Then $\operatorname{id}_{\mathbf C}$ is a functor.
Category of Categories is Category
Let $\mathbf{Cat}$ be the category of categories. Then $\mathbf{Cat}$ is a metacategory.
Identity Functor is Left Identity
Let $\mathbf C$ and $\mathbf D$ be metacategories. Let $F: \mathbf C \to \mathbf D$ be a functor, and let $\operatorname{id}_{\mathbf D}$ be the identity functor on $\mathbf D$. Then the composite functor $\operatorname{id}_{\mathbf D} F$ satisfies: :$\operatorname{id}_{\mathbf D} F = F$
Identity Functor is Right Identity
Let $\mathbf C$ and $\mathbf D$ be metacategories. Let $F: \mathbf C \to \mathbf D$ be a functor, and let $\operatorname{id}_{\mathbf C}$ be the identity functor on $\mathbf C$. Then the composite functor $F \operatorname{id}_{\mathbf C}$ satisfies: :$F \operatorname{id}_{\mathbf C} = F$
Composition of Functors is Associative
Let $\mathbf A$, $\mathbf B$, $\mathbf C$ and $\mathbf D$ be metacategories. Let $F: \mathbf A \to \mathbf B$, $G: \mathbf B \to \mathbf C$ and $H: \mathbf C \to \mathbf D$ be functors. Composition of functors is associative: :$H \paren {G F} = \paren {H G} F$
Abel-Ruffini Theorem
There is no general algebraic solution for determining all the roots of a polynomial of degree $5$ or higher.
Cartesian Product is Small
Let $a$ and $b$ be small classes. Then their Cartesian product $\left({a \times b}\right)$ is small: :$\mathscr M \left({a \times b}\right)$
Preorder Category is Category
Let $\struct {S, \precsim}$ be a preordered set. Let $\mathbf S$ be its associated preorder category. Then $\mathbf S$ is a category.
Category Induces Preorder
Let $\mathbf S$ be a category with set of objects $S$. Then the binary relation $\precsim$ defined by: :$\forall a, b \in S: a \precsim b \iff \exists f: a \to b$ is a preorder on $S$.
Preorder Induced by Preorder Category
Let $\left({S, \precsim}\right)$ be a preordered set. Let $\mathbf S$ be its associated preorder category. Let $\precsim'$ be the preorder induced by $\mathbf S$ as on Category Induces Preorder. Then $\precsim'$ is the same as $\precsim$.
Functor between Order Categories
Let $\struct {S, \preceq}$ and $\struct {T, \preceq'}$ be ordered sets. Let $\mathbf S$ and $\mathbf T$ be their associated order categories, respectively. Let $F: \mathbf S \to \mathbf T$ be a functor. Then its object functor $F: S \to T$ is a monotone mapping.
Discrete Category on Set is Discrete Category
Let $S$ be a set. Let $\mathbf{Dis} \left({S}\right)$ be the discrete category on $S$. Then $\mathbf{Dis} \left({S}\right)$ determines a unique (up to isomorphism discrete category $\mathbf{Dis} \left({S}\right)$ whose objects precisely comprise $S$. {{expand|That is, $\mathbf{Dis}: \mathbf{Set} \to \mathbf{Cat}$ is a functor (with left inverse $\mathbf{ob}$) }}
Image of Small Class under Mapping is Small
Let $A$ be a mapping. Let $a$ be a small class. Then, the image of $a$ under $A$ is small. {{improve|$A$ being a class variable, I strongly suggest a different letter to denote the mapping}}
Inverse of Small Relation is Small
Let $a$ be a small class. Let $a$ also be a relation. Then the inverse relation of $a$ is small.
Functor between Monoid Categories
Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be monoids. Let $\mathbf S$ and $\mathbf T$ be the associated monoid categories. Let $F: \mathbf S \to \mathbf T$ be a functor. Then the morphism functor $F_1$ of $F$ is a monoid homomorphism.
Cayley's Theorem (Category Theory)
Let $\mathbf C$ be a small category. Denote with $\mathbf{Set}$ the category of sets. Then there exists a category $\mathbf D$, subject to: :$(1): \quad $ The objects of $\mathbf D$ are sets. :$(2): \quad $ The morphisms of $\mathbf D$ are mappings. :$(3): \quad \mathbf C \cong \mathbf D$, i.e. $\mathbf C$ and $\mathbf D$ are isomorphic. That is, $\mathbf C$ is isomorphic to a subcategory of $\mathbf{Set}$.
Domain of Small Relation is Small
Let $a$ be a small class. Let $a$ also be a relation. Then the domain of $a$ is small.
Range of Small Relation is Small
Let $a$ be a small class. Let $a$ also be a relation. Then the range of $a$ is small.
Definition:Opposite Group
Let $\struct {G, \circ}$ be a group. We define a new operation $*$ on $G$ by: :$\forall a, b \in G: a * b = b \circ a$ The algebraic structure $\struct {G, *}$ is called the '''opposite group''' to $G$.
Center of Opposite Group
Let $\struct {G, \circ}$ be a group. Let $\struct {G, *}$ be the opposite group to $G$. Let $\map Z {G, \circ}$ and $\map Z {G, *}$ be the centers of $\struct {G, \circ}$ and $\struct {G, *}$, respectively. Then: :$\map Z {G, \circ} = \map Z {G, *}$
Opposite Group of Opposite Group
Let $\left({G, \circ}\right)$ be a group. Let $\left({G, *}\right)$ be the opposite group to $\left({G, \circ}\right)$. Let $\left({G, \circ'}\right)$ be the opposite group to $\left({G, *}\right)$. Then: :$\left({G, \circ}\right) = \left({G, \circ'}\right)$
Slice Category is Category
Let $\mathbf C$ be a metacategory. Let $C \in \mathbf C_0$ be a object of $\mathbf C$. Let $\mathbf C / C$ be the slice category of $\mathbf C$ over $C$. Then $\mathbf C / C$ is a metacategory.
Monoid Category is Category
Let $\left({S, \circ}\right)$ be a monoid with identity $e_S$. Let $\mathbf S$ be the associated monoid category. Then $\mathbf S$ is a category.
Dual Category is Category
Let $\mathbf C$ be a metacategory. Let $\mathbf C^{\text{op}}$ be its dual category. Then $\mathbf C^{\text{op}}$ is also a metacategory.
Signum Function on Integers is Extension of Signum on Natural Numbers
Let $\sgn_\Z: \Z \to \set {-1, 0, 1}$ be the signum function on the integers. Let $\sgn_\N: \N \to \set {0, 1}$ be the signum function on the natural numbers. Then $\sgn_\Z: \Z \to \Z$ is an extension of $\sgn_\N: \N \to \N$.
Cartesian Product with Proper Class is Proper Class
Let $A$ be a proper class. Let $B$ be a class which is not empty. Then the Cartesian product $\paren {A \times B}$ is a proper class.
Uniqueness Condition for Relation Value
Let $\mathcal R$ be a relation. Let $\left({x, y}\right) \in \mathcal R$. Let: :$\exists ! y: \left({x, y}\right) \in \mathcal R$ Then: :$\mathcal R \left({x}\right) = y$ where $\mathcal R \left({x}\right)$ denotes the image of $\mathcal R$ at $x$. If $y$ is not unique, then: :$\mathcal R \left({x}\right) = \varnothing$
Value of Relation is Small
The value of a relation is always a small class.
Mapping whose Domain is Small Class is Small
Let $F$ be a mapping. Let the domain of $F$ be a small class. Then, $F$ is a small class.
Restriction of Mapping to Small Class is Small
Let $F$ be a mapping. Let $A$ be a small class. Then the restriction $F {\restriction_A}$ is a small class.
Preimage of Singleton
Let $\RR$ be a relation. Let $\map {\RR^{-1} } t$ denote the preimage of $t$ under $\RR$. Let $\RR^{-1} \sqbrk {\set t}$ denote the preimage of $\set t$ under $\RR$. Then: :$\RR^{-1} \sqbrk {\set t} = \map {\RR^{-1} } t$
Order Isomorphism Preserves Minimal Elements
Let $A_1$ and $A_2$ be classes. Let $\prec_1$ and $\prec_2$ be relations. Let $\phi : A_1 \to A_2$ create an order isomorphism between $\left({ A_1 , \prec_1 }\right)$ and $\left({ A_2 , \prec_2 }\right)$. Let $B \subseteq A_1$. Then $\phi$ maps the $\prec_1$-minimal elements (for strict orderings) of $B$ to the $\prec_2$-minimal element of $\phi \left({B}\right)$.
Order Isomorphism Preserves Initial Segments
Let $A_1$ and $A_2$ be classes. Let $\prec_1$ and $\prec_2$ be strict orderings. Let $\phi : A_1 \to A_2$ create an order isomorphism between $\left({A_1, \prec_1}\right)$ and $\left({A_2, \prec_2}\right)$. Suppose $x \in A_1$. Then $\phi$ maps the $\prec_1$-initial segment of $x$ to the $\prec_2$-initial segment of $\phi \left({x}\right)$.
Alternating Group is Set of Even Permutations
Let $S_n$ denote the symmetric group on $n$ letters. Let $A_n$ be the alternating group on $n$ letters. Then $A_n$ consists of the set of even permutations of $S_n$.
Induced Relation Generates Order Isomorphism
Let $\left({ A_1 , \preceq_1 }\right)$ be an ordered set. Let $\phi : A_1 \to A_2$ be a bijection. Let: : $\preceq_2 \mathop{:=} \left\{ \left({\phi\left({x}\right), \phi\left({y}\right) }\right): x \in A_1 \land y \in A_1 \land x \mathop{\preceq_1} y \right\}$ Then $\phi : \left({ A_1 , \preceq_1 }\right) \to \left({ A_2 , \preceq_2 }\right)$ is an order isomorphism.
Ordinal Multiplication is Left Distributive
Let $x$, $y$, and $z$ be ordinals. Let $\times$ denote ordinal multiplication. Let $+$ denote ordinal addition. Then: :$x \times \left({ y + z }\right) = \left({ x \times y }\right) + \left({ x \times z }\right)$
Ordinal Multiplication is Associative
Let $x, y, z$ be ordinals. Let $\times$ denote ordinal multiplication. Then: :$x \times \left({y \times z}\right) = \left({x \times y}\right) \times z$
Division Theorem for Ordinals
Let $x$ and $y$ be ordinals. Let $0$ denote the zero ordinal. Suppose $y \ne 0$. Then there exist unique ordinals $z$ and $w$ such that: :$x = \paren {y \times z} + w$ and $w < y$.
Finite Ordinal Times Ordinal
Let $m$ and $n$ be finite ordinals. Let $m \ne 0$, where $0$ is the zero ordinal. Let $x$ be a limit ordinal. Then: :$m \times \left({ x + n }\right) = x + \left({ m \times n }\right)$ {{expand|Via Cantor normal form, all ordinals are of the form $x+n$}}
Factorization of Limit Ordinals
Let $x$ be a limit ordinal. Then: :$x = ( \omega \times y )$ for some $y \in \operatorname{On}$ where $\omega$ is the minimal infinite successor set.
Ordinal Exponentiation is Closed
Let $x$ and $y$ be ordinals. Then: :$x^y \in \operatorname{On}$ That is, ordinal exponentiation is closed.
Exponent Base of One
Let $x$ be an ordinal. Then: :$1^x = 1$
Exponent Not Equal to Zero
Let $x$ and $y$ be ordinals. Let $x \ne 0$. Then: :$x^y \ne 0$
Membership is Left Compatible with Ordinal Exponentiation
Let $x$, $y$, and $z$ be ordinals. Suppose $1 < z$. Then: :$x < y \iff z^x < z^y$
Conjugacy Action on Group Elements is Group Action
The conjugacy action on $G$: : $\forall g, h \in G: g * h = g \circ h \circ g^{-1}$ is a  group action on itself.
Conjugacy Action on Subgroups is Group Action
Let $X$ be the set of all subgroups of $G$. For any $H \le G$ and for any $g \in G$, the conjugacy action: :$g * H := g \circ H \circ g^{-1}$ is a group action.
Slice Category of Order Category
Let $\mathbf P$ be a order category, and denote its ordering by $\preceq$. Let $p \in \mathbf P_0$ be an object of $\mathbf P$. Then: :$\mathbf P \mathbin / p \cong p^\preceq$ where: :$\mathbf P \mathbin / p$ is the slice of $\mathbf P$ over $p$ :$p^\preceq$ is the order category defined by the weak lower closure of $p$.
Category of Pointed Sets as Coslice Category
Let $\mathbf{Set}_*$ be the category of pointed sets. Let $\mathbf{Set}$ be the category of sets. Let $1 := \left\{{*}\right\}$ be any singleton. Then: :$\mathbf{Set}_* \cong 1 \mathbin / \mathbf{Set}$ where $1 \mathbin / \mathbf{Set}$ denotes the coslice of $\mathbf{Set}$ under $1$ and $\cong$ signifies isomorphic categories.
Subset is Right Compatible with Ordinal Exponentiation
Let $x, y, z$ be ordinals. Then: :$x \le y \implies x^z \le y^z$
Condition for Membership is Right Compatible with Ordinal Exponentiation
Let $x, y, z$ be ordinals. Let $z$ be the successor of some ordinal $w$. Then: :$x < y \iff x^z < y^z$
Lower Bound for Ordinal Exponentiation
Let $x$ and $y$ be ordinals. Let $x$ be greater than $1$, where $1$ denotes the successor of the zero ordinal. Then: :$y \le x^y$
Unique Ordinal Exponentiation Inequality
Let $x$ and $y$ be ordinals. Let $x > 1$ and $y > 0$. Then there exists a unique ordinal $z$ such that: :$x^z ≤ y$ and $y < x^{z^+}$
Limit Ordinals Closed under Ordinal Exponentiation
Let $x$ and $y$ be ordinals. Let $y$ be a limit ordinal. Let $x^y$ denote ordinal exponentiation. Then: :If $x > 1$, then $x^y$ is a limit ordinal. :If $x \ne \varnothing$, then $y^x$ is a limit ordinal.
Successor of Element of Ordinal is Subset
Let $x$ and $y$ be ordinals. Then: :$x \in y \iff x^+ \subseteq y$
Ordinal is Less than Ordinal to Limit Power
Let $x$, $y$, and $z$ be ordinals. Let $z$ be a limit ordinal. Then: :$x < y^z \iff \exists w \in z: x < y^w$
Ordinal Sum of Powers
Let $x$, $y$, and $z$ be ordinals. Then: :$x^y \times x^z = x^{y + z}$
Kleene Closure is Monoid
Let $S$ be a set, and let $S^*$ be its Kleene closure. Let $*$ denote concatenation of ordered tuples. Then $\struct {S^*, *}$ is a monoid.
Ordinal Power of Power
Let $x$, $y$, and $z$ be ordinals. Then: :$\left({x^y}\right)^z = x^{y \mathop \times z}$
Upper Bound of Ordinal Sum
Let $x$ and $y$ be ordinals. Suppose $x > 1$. Let $\langle a_n \rangle$ be a finite sequence of ordinals such that: :$a_n < x$ for all $n$ Let $\left\langle{b_n}\right\rangle$ be a strictly decreasing finite sequence of ordinals such that: :$b_n < y$ for all $n$ Then: :$\displaystyle \sum_{i \mathop = 1}^n x^{b_i} a_i < x^y$
General Associative Law for Ordinal Sum
Let $x$ be a finite ordinal. Let $\left\langle{a_i}\right\rangle$ be a sequence of ordinals. Then: :$\displaystyle \sum_{i \mathop = 1}^{x + 1} a_i = a_1 + \sum_{i \mathop = 1}^x a_{i + 1}$
Power Sets of Equinumerous Sets are Equinumerous
Let $x$ and $y$ be sets such that $x \sim y$. Then: :$\powerset x \sim \powerset y$
General Associative Law for Ordinal Sum/Proof 1
Let $x$ be a finite ordinal. Let $\left\langle{a_i}\right\rangle$ be a sequence of ordinals. Then: :$\displaystyle \sum_{i \mathop = 1}^{x + 1} a_i = a_1 + \sum_{i \mathop = 1}^x a_{i + 1}$
General Associative Law for Ordinal Sum/Proof 2
Let $x$ be a finite ordinal. Let $\left\langle{a_i}\right\rangle$ be a sequence of ordinals. Then: :$\displaystyle \sum_{i \mathop = 1}^{x + 1} a_i = a_1 + \sum_{i \mathop = 1}^x a_{i + 1}$
Basis Representation Theorem for Ordinals
Let $x$ and $y$ be ordinals. Let $x > 1$ and $y > 0$. Then there exist unique finite sequences of ordinals: :$\sequence {a_i}, \sequence {b_i}$ both of unique length $n$ such that: :$(1): \quad \sequence {a_i}$ is a strictly decreasing sequence for $1 \le i \le n$ :$(2): \quad 0 < b_i < x$ for all $1 \le i \le n$ :$(3): \quad \ds y = \sum_{i \mathop = 1}^n x^{a_i} b_i$
Ordinal Multiplication via Cantor Normal Form/Limit Base
Let $x$ and $y$ be ordinals. Let $x$ be a limit ordinal. Let $y > 0$. Let $\left\langle{a_i}\right\rangle$ be a sequence of ordinals that is strictly decreasing on $1 \le i \le n$. Let $\left\langle{b_i}\right\rangle$ be a sequence of finite ordinals. Then: :$\displaystyle \sum_{i \mathop = 1}^n \left({x^{a_i} b_i}\right) \times x^y = x^{a_1 \mathop + y}$
Ordinal Exponentiation of Terms
Let $x, y, z$ be ordinals. Let $n$ be a finite ordinal. Let $x$ be a limit ordinal. Let $y, z, n$ all be greater than $0$. Then: :$\left({x^y \times n}\right)^z = x^{y \mathop \times z} \times n$ if $z$ is not a limit ordinal :$\left({x^y \times n }\right)^z = x^{y \mathop \times z}$ if $z$ is a limit ordinal
Prime Triplet is Unique
Let $n \in \Z$ be an integer such that $n > 3$. Then $n$, $n + 2$, $n + 4$ cannot all be prime. That is, the only prime triplet is $\set {3, 5, 7}$.
Ordinal Exponentiation via Cantor Normal Form/Limit Exponents
Let $x$ and $y$ be ordinals. Let $x$ and $y$ be limit ordinals. Let $\sequence {a_i}$ be a sequence of ordinals that is strictly decreasing on $1 \le i \le n$. Let $\sequence {b_i}$ be a sequence of natural numbers. Then: :$\displaystyle \paren {\sum_{i \mathop = 1}^n x^{a_i} \times b_i}^y = x^{a_1 \mathop \times y}$
Inverse Relation Functor is Contravariant Functor
Let $\mathbf{Rel}$ be the category of relations. Let $C: \mathbf{Rel} \to \mathbf{Rel}$ be the inverse relation functor. Then $C$ is a contravariant functor.
Injection iff Monomorphism in Category of Sets
Let $\mathbf{Set}$ be the category of sets. Let $f: X \to Y$ be a morphism in $\mathbf{Set}$, i.e. a mapping. Then $f$ is an injection iff it is a monomorphism.
Morphism in Preorder Category is Monic
Let $\mathbf P$ be a preorder category. Let $f \in \mathbf P_1$ be a morphism. Then $f$ is monic.
Surjection iff Epimorphism in Category of Sets
Let $\mathbf{Set}$ be the category of sets. Let $f: X \to Y$ be a morphism in $\mathbf{Set}$, i.e. a mapping. Then $f$ is a surjection iff it is an epimorphism.
Inclusion of Natural Numbers in Integers is Epimorphism
Let $\mathbf{Mon}$ be the category of monoids. Let $\left({\N, +}\right)$ denote the monoid of natural numbers as on Natural Numbers under Addition form Commutative Monoid. Let $\left({\Z, +}\right)$ denote the monoid of integers as on additive group of integers. Denote with $\iota: \N \to \Z$ the inclusion mapping. Then $\iota: \N \to \Z$ is an epimorphism in $\mathbf{Mon}$.
Isomorphism (Category Theory) is Monic
Let $\mathbf C$ be a metacategory. Let $f: C \to D$ be an isomorphism. Then $f: C \rightarrowtail D$ is monic.
Stabilizer of Element under Conjugacy Action is Centralizer
Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $*$ be the conjugacy action on $G$ defined by the rule: : $\forall g, h \in G: g * h = g \circ h \circ g^{-1}$ Let $x \in G$. Then the stabilizer of $x$ under this conjugacy action is: :$\Stab x = \map {C_G} x$ where $\map {C_G} x$ is the centralizer of $x$ in $G$.
Stabilizer of Subset Product Action on Power Set
Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $\powerset G$ be the power set of $\struct {G, \circ}$. Let $*: G \times \powerset G \to \powerset G$ be the subset product action on $\powerset G$ defined as: :$\forall g \in G: \forall S \in \powerset G: g * S = g \circ S$ where $g \circ S$ is the subset product $\set g \circ S$. Then the stabilizer of $S$ in $\powerset G$ is the set: :$\Stab S = S$
Stabilizer of Coset Action on Set of Subgroups
Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $\powerset G$ denote the power set of $G$. Let $\HH \subseteq \powerset G$ denote the set of subgroups of $G$. Let $*$ be the subset product action on $\HH \subseteq \powerset G$ defined as: :$\forall g \in G: \forall H \in \HH: g * H = g \circ H$ where $g \circ H$ is the (left) coset of $g$ by $H$. Then the stabilizer of $H$ in $\powerset G$ is $H$ itself: :$\Stab H = H$
Isomorphism (Category Theory) is Epic
Let $\mathbf C$ be a metacategory. Let $f: C \to D$ be an isomorphism. Then $f: C \twoheadrightarrow D$ is epic.
Split Epimorphism is Epic
Let $\mathbf C$ be a metacategory. Let $f: C \to D$ be a split epimorphism. Then $f: C \twoheadrightarrow D$ is epic.
Split Monomorphism is Monic
Let $\mathbf C$ be a metacategory. Let $f: C \to D$ be a split monomorphism. Then $f: C \rightarrowtail D$ is monic.
Relational Closure Exists for Set-Like Relation
Let $A$ be a class. Let $\prec$ be a relation on $A$. Furthermore, let ${\prec^{-1}}\left({a}\right)$ be a small class for each $a \in A$. Let $S$ be a small class that is a subset of the class $A$. Let $G$ be a mapping such that: :$G \left({ x }\right) = A \cap \left({ {\prec^{-1}} \left({ x }\right) }\right)$ Let $F$ be defined using the Principle of Recursive Definition: :$F \left({0}\right) = S$ :$F \left({n^+}\right) = F \left({n}\right) \cup G \left({F \left({n}\right)}\right)$ Let $\displaystyle T = \bigcup_{n \mathop \in \omega} F\left({n}\right)$. Then: :$(1): \quad T$ is a set and satisfies: :::::$\forall x \in A: \forall y \in T: \left({ x \prec y \implies x \in T }\right)$ :::::In other words, $T$ is $\prec$-transitive. :$(2): \quad S \subseteq T$ :$(3): \quad$ If $R$ is $\prec$-transitive and $S \subseteq R$, then $T \subseteq R$. That is, given any set $S$, there is an explicit construction for its relational closure.
Stabilizer of Conjugacy Action on Subgroup is Normalizer
Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $X$ be the set of all subgroups of $G$. Let $*$ be the conjugacy action on $H$: :$\forall g \in G, H \in X: g * H = g \circ H \circ g^{-1}$ Then the stabilizer of $H$ in $\powerset G$ is given by: :$\Stab H = \map {N_G} H$ where $\map {N_G} H$ is the normalizer of $H$ in $G$.
Well-Founded Proper Relational Structure Determines Minimal Elements
Let $A$ and $B$ be classes. Let $\left({ A, \prec }\right)$ be a proper relational structure. Let $\prec$ be a foundational relation. Suppose $B \subset A$ and $B \ne \varnothing$. Then $B$ has a $\prec$-minimal element.
Initial Object is Unique
Let $\mathbf C$ be a metacategory. Let $0$ and $0'$ be two initial objects of $\mathbf C$. Then there is a unique isomorphism $u: 0 \to 0'$. Hence, initial objects are unique up to unique isomorphism.
Terminal Object is Unique
Let $\mathbf C$ be a metacategory. Let $1$ and $1'$ be two terminal objects of $\mathbf C$. Then there is a unique isomorphism $u: 1 \to 1'$. Hence, terminal objects are unique up to unique isomorphism.
Empty Set is Initial Object
Let $\mathbf{Set}$ be the category of sets. Then the empty set $\O$ is an initial object of $\mathbf{Set}$.
Singleton is Terminal Object of Category of Sets
Let $\mathbf{Set}$ be the category of sets. Let $S = \left\{{x}\right\}$ be any singleton set. Then $S$ is a terminal object of $\mathbf{Set}$.
Zero (Category) is Initial Object
Let $\mathbf{Cat}$ be the category of categories. Let $\mathbf 0$ be the zero category. Then $\mathbf 0$ is an initial object of $\mathbf{Cat}$.
One (Category) is Terminal Object
Let $\mathbf{Cat}$ be the category of categories. Let $\mathbf 1$ be the category one. Then $\mathbf 1$ is a terminal object of $\mathbf{Cat}$.
Empty Mapping is Mapping
For each set $T$, the empty mapping, where the domain is the empty set, is a mapping.
Empty Mapping is Unique
For each set $T$ there exists exactly one empty mapping $f: \O \to T$ whose domain is the empty set.
Null Relation is Mapping iff Domain is Empty Set
Let $S$ and $T$ be sets. The null relation $\RR = \O \subseteq S \times T$ is a mapping  $S = \O$.
Orbit of Element of Group Acting on Itself is Group
Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $*$ be the group action of $\struct {G, \circ}$ on itself by the rule: :$\forall g, h \in G: g * h = g \circ h$ Then the orbit of an element $x \in G$ is given by: :$\Orb x = G$
Orbit of Element under Conjugacy Action is Conjugacy Class
Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $*$ be the conjugacy group action on $G$: : $\forall g, h \in G: g * h = g \circ h \circ g^{-1}$ Let $x \in G$. Then the orbit of $x$ under this group action is: :$\Orb x = C_x$ where $C_x$ is the conjugacy class of $x$.
Axiom of Foundation (Strong Form)
Let $B$ be a class. Suppose $B$ is non-empty. Then $B$ has a $\in$-minimal element.
Trivial Group is Terminal Object of Category of Groups
Let $\mathbf{Grp}$ be the category of groups. Let $\left\{{e}\right\}$ be the trivial group. Then $\left\{{e}\right\}$ is a terminal object of $\mathbf{Grp}$.
Orbit of Conjugacy Action on Subgroup is Set of Conjugate Subgroups
Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $X$ be the set of all subgroups of $G$. Let $*$ be the conjugacy action on $H$ defined as: : $\forall g \in G, H \in X: g * H = g \circ H \circ g^{-1}$ Then the orbit $\Orb H$ of $H$ in $\powerset G$ is the set of subgroups of $G$ conjugate to $H$.
Closure for Finite Collection of Relations and Operations
Let $\RR_1, \RR_2, \ldots \RR_n$ be relations. Let $\SS_1, \SS_2, \ldots \SS_m$ be operations. Let $T$ be a small class. Let the image of $\RR_i$ over any small class $x$ be small classes for $1 \le i \le n$. Let the image of $\SS_i$ over any Cartesian product $x \times x$ be small classes for $1 \le i \le m$. Then there exists a small class $X$ such that: :$(1): \quad$ $T \subseteq X$ :$(2): \quad$ Each $\RR_i$ is closed with respect to $X$.  Each $\SS_i$ is closed with respect to $X \times X$. :$(3): \quad$ $X$ is the smallest small class satisfying conditions $(1)$ and $(2)$ above. If $Y$ satisfies conditions $(1)$ and $(2)$, then $X \subseteq Y$.