Theorem 4cycl2v2nb 28062

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 4748	. 2 ⊢ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → {{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸)
2		prcom 4662	. . . . 5 ⊢ {𝐷, 𝐴} = {𝐴, 𝐷}
3	2	eleq1i 2903	. . . 4 ⊢ ({𝐷, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐷} ∈ 𝐸)
4	3	biimpi 218	. . 3 ⊢ ({𝐷, 𝐴} ∈ 𝐸 → {𝐴, 𝐷} ∈ 𝐸)
5		prcom 4662	. . . . 5 ⊢ {𝐶, 𝐷} = {𝐷, 𝐶}
6	5	eleq1i 2903	. . . 4 ⊢ ({𝐶, 𝐷} ∈ 𝐸 ↔ {𝐷, 𝐶} ∈ 𝐸)
7	6	biimpi 218	. . 3 ⊢ ({𝐶, 𝐷} ∈ 𝐸 → {𝐷, 𝐶} ∈ 𝐸)
8		prssi 4748	. . 3 ⊢ (({𝐴, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐶} ∈ 𝐸) → {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸)
9	4, 7, 8	syl2anr 598	. 2 ⊢ (({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸) → {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸)
10	1, 9	anim12i 614	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 398   ∈ wcel 2110   ⊆ wss 3936  {cpr 4563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3497  df-un 3941  df-in 3943  df-ss 3952  df-sn 4562  df-pr 4564

This theorem is referenced by:  4cycl2vnunb  28063