-- Module: Mathlib.Combinatorics.SimpleGraph.Tutte
Goal:
V : Type u_1
G : SimpleGraph V
inst : Finite V
x a b c : V
M1 : (G ⊔ edge x b).Subgraph
M2 : (G ⊔ edge a c).Subgraph
hxa : G.Adj x a
hab : G.Adj a b
hnGxb : ¬G.Adj x b
hnGac : ¬G.Adj a c
hnxb : x ≠ b
hnxc : x ≠ c
hnac : a ≠ c
hnbc : b ≠ c
hM1 : M1.IsPerfectMatching
hM2 : M2.IsPerfectMatching
hM1xb : M1.Adj x b
hM2ac : M2.Adj a c
hM1nac : ¬M1.Adj a c
hsupG : G ⊔ edge x b ⊔ (G ⊔ edge a c) = G ⊔ edge a c ⊔ edge x b
cycles : SimpleGraph V := M1.spanningCoe ∆ M2.spanningCoe
hcalt : cycles.IsAlternating M2.spanningCoe
hcycles : (M1.spanningCoe ∆ M2.spanningCoe).IsCycles
hcac : cycles.Adj a c
hM1sub : M1.spanningCoe ≤ G ⊔ edge x b
hM2sub : M2.spanningCoe ≤ G ⊔ edge a c
induce_le :
  (cycles.connectedComponentMk c).toSimpleGraph.spanningCoe (s := (↑(cycles.connectedComponentMk c) : Set V)) ≤
    G ⊔ edge a c ⊔ edge x b
hxc : x ∈ (cycles.connectedComponentMk c).supp
hacc : a ∈ (cycles.connectedComponentMk c).supp
this_1 : (G : SimpleGraph V) → G.LocallyFinite
hnM2 : ∀ (x' : V) (h : x' ≠ c), ¬M2.Adj x' a
p : (M1.spanningCoe ∆ M2.spanningCoe).Walk a a
hp : p.IsCycle ∧ p.toSubgraph.verts = (cycles.connectedComponentMk c).supp
p' : (M1.spanningCoe ∆ M2.spanningCoe).Walk a a
hp' : p'.IsCycle ∧ p'.snd = c ∧ p'.toSubgraph.verts = p.toSubgraph.verts
x' : V
hx'p : x' ∈ p'.support
htw : ∀ (t : V), Membership.mem (γ := Finset V) {x, b} t → t ∈ (p'.takeUntil x' hx'p).support → t = x'
this : (p'.takeUntil x' hx'p).toSubgraph.Adj a (p'.takeUntil x' hx'p).snd
hx' : x' = x ∨ x' = b
refine_1 :
  ∀ {M1 : (G ⊔ edge x' b).Subgraph} (hxa : G.Adj x' a) (hnGxb : ¬G.Adj x' b) (hnxb : x' ≠ b) (hnxc : x' ≠ c)
    (hM1 : M1.IsPerfectMatching) (hM1xb : M1.Adj x' b) (hM1nac : ¬M1.Adj a c)
    (hsupG : G ⊔ edge x' b ⊔ (G ⊔ edge a c) = G ⊔ edge a c ⊔ edge x' b),
    let cycles := M1.spanningCoe ∆ M2.spanningCoe;
    ∀ (hcalt : cycles.IsAlternating M2.spanningCoe) (hcycles : (M1.spanningCoe ∆ M2.spanningCoe).IsCycles)
      (hcac : cycles.Adj a c) (hM1sub : M1.spanningCoe ≤ G ⊔ edge x' b)
      (induce_le :
        (cycles.connectedComponentMk c).toSimpleGraph.spanningCoe (s := (↑(cycles.connectedComponentMk c) : Set V)) ≤
          G ⊔ edge a c ⊔ edge x' b)
      (hxc : x' ∈ (cycles.connectedComponentMk c).supp) (hacc : a ∈ (cycles.connectedComponentMk c).supp)
      (p : (M1.spanningCoe ∆ M2.spanningCoe).Walk a a)
      (hp : p.IsCycle ∧ p.toSubgraph.verts = (cycles.connectedComponentMk c).supp)
      (p' : (M1.spanningCoe ∆ M2.spanningCoe).Walk a a)
      (hp' : p'.IsCycle ∧ p'.snd = c ∧ p'.toSubgraph.verts = p.toSubgraph.verts) (hx'p : x' ∈ p'.support)
      (htw : ∀ (t : V), Membership.mem (γ := Finset V) {x', b} t → t ∈ (p'.takeUntil x' hx'p).support → t = x')
      (this : (p'.takeUntil x' hx'p).toSubgraph.Adj a (p'.takeUntil x' hx'p).snd), x' ≠ a
refine_1₁ :
  ∀ {M1 : (G ⊔ edge x x').Subgraph} (hab : G.Adj a x') (hnGxb : ¬G.Adj x x') (hnxb : x ≠ x') (hnbc : x' ≠ c)
    (hM1 : M1.IsPerfectMatching) (hM1xb : M1.Adj x x') (hM1nac : ¬M1.Adj a c)
    (hsupG : G ⊔ edge x x' ⊔ (G ⊔ edge a c) = G ⊔ edge a c ⊔ edge x x'),
    let cycles := M1.spanningCoe ∆ M2.spanningCoe;
    ∀ (hcalt : cycles.IsAlternating M2.spanningCoe) (hcycles : (M1.spanningCoe ∆ M2.spanningCoe).IsCycles)
      (hcac : cycles.Adj a c) (hM1sub : M1.spanningCoe ≤ G ⊔ edge x x')
      (induce_le :
        (cycles.connectedComponentMk c).toSimpleGraph.spanningCoe (s := (↑(cycles.connectedComponentMk c) : Set V)) ≤
          G ⊔ edge a c ⊔ edge x x')
      (hxc : x ∈ (cycles.connectedComponentMk c).supp) (hacc : a ∈ (cycles.connectedComponentMk c).supp)
      (p : (M1.spanningCoe ∆ M2.spanningCoe).Walk a a)
      (hp : p.IsCycle ∧ p.toSubgraph.verts = (cycles.connectedComponentMk c).supp)
      (p' : (M1.spanningCoe ∆ M2.spanningCoe).Walk a a)
      (hp' : p'.IsCycle ∧ p'.snd = c ∧ p'.toSubgraph.verts = p.toSubgraph.verts) (hx'p : x' ∈ p'.support)
      (htw : ∀ (t : V), Membership.mem (γ := Finset V) {x, x'} t → t ∈ (p'.takeUntil x' hx'p).support → t = x')
      (this : (p'.takeUntil x' hx'p).toSubgraph.Adj a (p'.takeUntil x' hx'p).snd), x' ≠ a
⊢ x' ≠ a

Tactic:
obtain rfl | rfl := hx'

Premises used: rfl