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Theorem 4cycl2vnunb 28119
 Description: In a 4-cycle, two distinct vertices have not a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Nov-2017.) (Revised by AV, 2-Apr-2021.)
Assertion
Ref Expression
4cycl2vnunb ((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) → ¬ ∃!𝑥𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐸   𝑥,𝑉
Allowed substitution hint:   𝐷(𝑥)

Proof of Theorem 4cycl2vnunb
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 4cycl2v2nb 28118 . 2 ((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸)) → ({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸))
2 preq2 4633 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐵 → {𝐴, 𝑥} = {𝐴, 𝐵})
3 preq1 4632 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐵 → {𝑥, 𝐶} = {𝐵, 𝐶})
42, 3preq12d 4640 . . . . . . . . . . . . . . 15 (𝑥 = 𝐵 → {{𝐴, 𝑥}, {𝑥, 𝐶}} = {{𝐴, 𝐵}, {𝐵, 𝐶}})
54sseq1d 3948 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → ({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ↔ {{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸))
65anbi1d 632 . . . . . . . . . . . . 13 (𝑥 = 𝐵 → (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ↔ ({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸)))
7 neeq1 3049 . . . . . . . . . . . . 13 (𝑥 = 𝐵 → (𝑥𝑦𝐵𝑦))
86, 7anbi12d 633 . . . . . . . . . . . 12 (𝑥 = 𝐵 → ((({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥𝑦) ↔ (({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝐵𝑦)))
9 preq2 4633 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐷 → {𝐴, 𝑦} = {𝐴, 𝐷})
10 preq1 4632 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐷 → {𝑦, 𝐶} = {𝐷, 𝐶})
119, 10preq12d 4640 . . . . . . . . . . . . . . 15 (𝑦 = 𝐷 → {{𝐴, 𝑦}, {𝑦, 𝐶}} = {{𝐴, 𝐷}, {𝐷, 𝐶}})
1211sseq1d 3948 . . . . . . . . . . . . . 14 (𝑦 = 𝐷 → ({{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸 ↔ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸))
1312anbi2d 631 . . . . . . . . . . . . 13 (𝑦 = 𝐷 → (({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ↔ ({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸)))
14 neeq2 3050 . . . . . . . . . . . . 13 (𝑦 = 𝐷 → (𝐵𝑦𝐵𝐷))
1513, 14anbi12d 633 . . . . . . . . . . . 12 (𝑦 = 𝐷 → ((({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝐵𝑦) ↔ (({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸) ∧ 𝐵𝐷)))
168, 15rspc2ev 3584 . . . . . . . . . . 11 ((𝐵𝑉𝐷𝑉 ∧ (({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸) ∧ 𝐵𝐷)) → ∃𝑥𝑉𝑦𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥𝑦))
17163expa 1115 . . . . . . . . . 10 (((𝐵𝑉𝐷𝑉) ∧ (({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸) ∧ 𝐵𝐷)) → ∃𝑥𝑉𝑦𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥𝑦))
1817expcom 417 . . . . . . . . 9 ((({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸) ∧ 𝐵𝐷) → ((𝐵𝑉𝐷𝑉) → ∃𝑥𝑉𝑦𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥𝑦)))
1918ex 416 . . . . . . . 8 (({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸) → (𝐵𝐷 → ((𝐵𝑉𝐷𝑉) → ∃𝑥𝑉𝑦𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥𝑦))))
2019com13 88 . . . . . . 7 ((𝐵𝑉𝐷𝑉) → (𝐵𝐷 → (({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸) → ∃𝑥𝑉𝑦𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥𝑦))))
21203impia 1114 . . . . . 6 ((𝐵𝑉𝐷𝑉𝐵𝐷) → (({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸) → ∃𝑥𝑉𝑦𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥𝑦)))
2221impcom 411 . . . . 5 ((({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) → ∃𝑥𝑉𝑦𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥𝑦))
23 rexnal 3201 . . . . . 6 (∃𝑥𝑉 ¬ ∀𝑦𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦) ↔ ¬ ∀𝑥𝑉𝑦𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦))
24 rexnal 3201 . . . . . . . 8 (∃𝑦𝑉 ¬ (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦) ↔ ¬ ∀𝑦𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦))
25 annim 407 . . . . . . . . . 10 ((({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ ¬ 𝑥 = 𝑦) ↔ ¬ (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦))
26 df-ne 2988 . . . . . . . . . . . 12 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
2726bicomi 227 . . . . . . . . . . 11 𝑥 = 𝑦𝑥𝑦)
2827anbi2i 625 . . . . . . . . . 10 ((({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ ¬ 𝑥 = 𝑦) ↔ (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥𝑦))
2925, 28bitr3i 280 . . . . . . . . 9 (¬ (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦) ↔ (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥𝑦))
3029rexbii 3211 . . . . . . . 8 (∃𝑦𝑉 ¬ (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦) ↔ ∃𝑦𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥𝑦))
3124, 30bitr3i 280 . . . . . . 7 (¬ ∀𝑦𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦) ↔ ∃𝑦𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥𝑦))
3231rexbii 3211 . . . . . 6 (∃𝑥𝑉 ¬ ∀𝑦𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦) ↔ ∃𝑥𝑉𝑦𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥𝑦))
3323, 32bitr3i 280 . . . . 5 (¬ ∀𝑥𝑉𝑦𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦) ↔ ∃𝑥𝑉𝑦𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥𝑦))
3422, 33sylibr 237 . . . 4 ((({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) → ¬ ∀𝑥𝑉𝑦𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦))
3534intnand 492 . . 3 ((({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) → ¬ (∃𝑥𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ ∀𝑥𝑉𝑦𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦)))
36 preq2 4633 . . . . . 6 (𝑥 = 𝑦 → {𝐴, 𝑥} = {𝐴, 𝑦})
37 preq1 4632 . . . . . 6 (𝑥 = 𝑦 → {𝑥, 𝐶} = {𝑦, 𝐶})
3836, 37preq12d 4640 . . . . 5 (𝑥 = 𝑦 → {{𝐴, 𝑥}, {𝑥, 𝐶}} = {{𝐴, 𝑦}, {𝑦, 𝐶}})
3938sseq1d 3948 . . . 4 (𝑥 = 𝑦 → ({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ↔ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸))
4039reu4 3672 . . 3 (∃!𝑥𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ↔ (∃𝑥𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ ∀𝑥𝑉𝑦𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦)))
4135, 40sylnibr 332 . 2 ((({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) → ¬ ∃!𝑥𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸)
421, 41stoic3 1778 1 ((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) → ¬ ∃!𝑥𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  ∃!wreu 3108   ⊆ wss 3883  {cpr 4530 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-v 3444  df-un 3888  df-in 3890  df-ss 3900  df-sn 4529  df-pr 4531 This theorem is referenced by:  n4cyclfrgr  28120  4cyclusnfrgr  28121
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