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Theorem 4cycl2v2nb-av 41459
 Description: In a (maybe degenerated) 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-av ((({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-av
StepHypRef Expression
1 prssi 4293 . 2 (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → {{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸)
2 prcom 4211 . . . . 5 {𝐷, 𝐴} = {𝐴, 𝐷}
32eleq1i 2679 . . . 4 ({𝐷, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐷} ∈ 𝐸)
43biimpi 205 . . 3 ({𝐷, 𝐴} ∈ 𝐸 → {𝐴, 𝐷} ∈ 𝐸)
5 prcom 4211 . . . . 5 {𝐶, 𝐷} = {𝐷, 𝐶}
65eleq1i 2679 . . . 4 ({𝐶, 𝐷} ∈ 𝐸 ↔ {𝐷, 𝐶} ∈ 𝐸)
76biimpi 205 . . 3 ({𝐶, 𝐷} ∈ 𝐸 → {𝐷, 𝐶} ∈ 𝐸)
8 prssi 4293 . . 3 (({𝐴, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐶} ∈ 𝐸) → {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸)
94, 7, 8syl2anr 494 . 2 (({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸) → {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸)
101, 9anim12i 588 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 383   ∈ wcel 1977   ⊆ wss 3540  {cpr 4127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-in 3547  df-ss 3554  df-sn 4126  df-pr 4128 This theorem is referenced by:  4cycl2vnunb-av  41460
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