Theorem 4cycl2v2nb 26543

Description: In a (maybe degenerated) 4-cycle, two vertices have two (maybe not different) common neighbors. (Contributed by Alexander van der Vekens, 19-Nov-2017.)

Assertion

Ref	Expression
4cycl2v2nb	⊢ ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) → ({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ ran 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ ran 𝐸))

Proof of Theorem 4cycl2v2nb

Step	Hyp	Ref	Expression
1		prssi 4293	. 2 ⊢ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → {{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ ran 𝐸)
2		prcom 4211	. . . . 5 ⊢ {𝐷, 𝐴} = {𝐴, 𝐷}
3	2	eleq1i 2679	. . . 4 ⊢ ({𝐷, 𝐴} ∈ ran 𝐸 ↔ {𝐴, 𝐷} ∈ ran 𝐸)
4	3	biimpi 205	. . 3 ⊢ ({𝐷, 𝐴} ∈ ran 𝐸 → {𝐴, 𝐷} ∈ ran 𝐸)
5		prcom 4211	. . . . 5 ⊢ {𝐶, 𝐷} = {𝐷, 𝐶}
6	5	eleq1i 2679	. . . 4 ⊢ ({𝐶, 𝐷} ∈ ran 𝐸 ↔ {𝐷, 𝐶} ∈ ran 𝐸)
7	6	biimpi 205	. . 3 ⊢ ({𝐶, 𝐷} ∈ ran 𝐸 → {𝐷, 𝐶} ∈ ran 𝐸)
8		prssi 4293	. . 3 ⊢ (({𝐴, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐶} ∈ ran 𝐸) → {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ ran 𝐸)
9	4, 7, 8	syl2anr 494	. 2 ⊢ (({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸) → {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ ran 𝐸)
10	1, 9	anim12i 588	1 ⊢ ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) → ({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ ran 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ ran 𝐸))

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977   ⊆ wss 3540  {cpr 4127  ran crn 5039

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  26544