Theorem 4cycl2v2nb 30051

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

Step	Hyp	Ref	Expression
1		prssi 4819	. 2 ⊢ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → {{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸)
2		prcom 4731	. . . . 5 ⊢ {𝐷, 𝐴} = {𝐴, 𝐷}
3	2	eleq1i 2818	. . . 4 ⊢ ({𝐷, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐷} ∈ 𝐸)
4	3	biimpi 215	. . 3 ⊢ ({𝐷, 𝐴} ∈ 𝐸 → {𝐴, 𝐷} ∈ 𝐸)
5		prcom 4731	. . . . 5 ⊢ {𝐶, 𝐷} = {𝐷, 𝐶}
6	5	eleq1i 2818	. . . 4 ⊢ ({𝐶, 𝐷} ∈ 𝐸 ↔ {𝐷, 𝐶} ∈ 𝐸)
7	6	biimpi 215	. . 3 ⊢ ({𝐶, 𝐷} ∈ 𝐸 → {𝐷, 𝐶} ∈ 𝐸)
8		prssi 4819	. . 3 ⊢ (({𝐴, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐶} ∈ 𝐸) → {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸)
9	4, 7, 8	syl2anr 596	. 2 ⊢ (({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸) → {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸)
10	1, 9	anim12i 612	1 ⊢ ((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸)) → ({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2098   ⊆ wss 3943  {cpr 4625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-un 3948  df-in 3950  df-ss 3960  df-sn 4624  df-pr 4626

This theorem is referenced by:  4cycl2vnunb  30052