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Theorem 4cycl2v2nb 28083
 Description: In a (maybe degenerate) 4-cycle, two vertice have two (maybe not different) common neighbors. (Contributed by Alexander van der Vekens, 19-Nov-2017.) (Revised by AV, 2-Apr-2021.)
Assertion
Ref Expression
4cycl2v2nb ((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸)) → ({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸))

Proof of Theorem 4cycl2v2nb
StepHypRef Expression
1 prssi 4738 . 2 (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → {{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸)
2 prcom 4653 . . . . 5 {𝐷, 𝐴} = {𝐴, 𝐷}
32eleq1i 2906 . . . 4 ({𝐷, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐷} ∈ 𝐸)
43biimpi 219 . . 3 ({𝐷, 𝐴} ∈ 𝐸 → {𝐴, 𝐷} ∈ 𝐸)
5 prcom 4653 . . . . 5 {𝐶, 𝐷} = {𝐷, 𝐶}
65eleq1i 2906 . . . 4 ({𝐶, 𝐷} ∈ 𝐸 ↔ {𝐷, 𝐶} ∈ 𝐸)
76biimpi 219 . . 3 ({𝐶, 𝐷} ∈ 𝐸 → {𝐷, 𝐶} ∈ 𝐸)
8 prssi 4738 . . 3 (({𝐴, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐶} ∈ 𝐸) → {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸)
94, 7, 8syl2anr 599 . 2 (({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸) → {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸)
101, 9anim12i 615 1 ((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸)) → ({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2115   ⊆ wss 3919  {cpr 4552 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553 This theorem is referenced by:  4cycl2vnunb  28084
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