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Theorem 4cycl2v2nb 27364
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 ((({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 4461 . 2 (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → {{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸)
2 prcom 4374 . . . . 5 {𝐷, 𝐴} = {𝐴, 𝐷}
32eleq1i 2794 . . . 4 ({𝐷, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐷} ∈ 𝐸)
43biimpi 206 . . 3 ({𝐷, 𝐴} ∈ 𝐸 → {𝐴, 𝐷} ∈ 𝐸)
5 prcom 4374 . . . . 5 {𝐶, 𝐷} = {𝐷, 𝐶}
65eleq1i 2794 . . . 4 ({𝐶, 𝐷} ∈ 𝐸 ↔ {𝐷, 𝐶} ∈ 𝐸)
76biimpi 206 . . 3 ({𝐶, 𝐷} ∈ 𝐸 → {𝐷, 𝐶} ∈ 𝐸)
8 prssi 4461 . . 3 (({𝐴, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐶} ∈ 𝐸) → {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸)
94, 7, 8syl2anr 496 . 2 (({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸) → {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸)
101, 9anim12i 591 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 2103  wss 3680  {cpr 4287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-v 3306  df-un 3685  df-in 3687  df-ss 3694  df-sn 4286  df-pr 4288
This theorem is referenced by:  4cycl2vnunb  27365
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