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Theorem 4cyclusnfrgr 29534
Description: A graph with a 4-cycle is not a friendhip graph. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 2-Apr-2021.)
Hypotheses
Ref Expression
4cyclusnfrgr.v 𝑉 = (Vtx‘𝐺)
4cyclusnfrgr.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
4cyclusnfrgr ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) → ((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸)) → 𝐺 ∉ FriendGraph ))

Proof of Theorem 4cyclusnfrgr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simprl 769 . . . . . 6 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) ∧ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸))) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))
2 simprr 771 . . . . . 6 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) ∧ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸))) → ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸))
3 simpl3 1193 . . . . . 6 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) ∧ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸))) → (𝐵𝑉𝐷𝑉𝐵𝐷))
4 4cycl2vnunb 29532 . . . . . 6 ((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) → ¬ ∃!𝑥𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸)
51, 2, 3, 4syl3anc 1371 . . . . 5 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) ∧ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸))) → ¬ ∃!𝑥𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸)
6 4cyclusnfrgr.v . . . . . . . . . 10 𝑉 = (Vtx‘𝐺)
7 4cyclusnfrgr.e . . . . . . . . . 10 𝐸 = (Edg‘𝐺)
86, 7frcond1 29508 . . . . . . . . 9 (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑥𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸))
9 pm2.24 124 . . . . . . . . 9 (∃!𝑥𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 → (¬ ∃!𝑥𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 → ¬ 𝐺 ∈ FriendGraph ))
108, 9syl6com 37 . . . . . . . 8 ((𝐴𝑉𝐶𝑉𝐴𝐶) → (𝐺 ∈ FriendGraph → (¬ ∃!𝑥𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 → ¬ 𝐺 ∈ FriendGraph )))
11103ad2ant2 1134 . . . . . . 7 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) → (𝐺 ∈ FriendGraph → (¬ ∃!𝑥𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 → ¬ 𝐺 ∈ FriendGraph )))
1211com23 86 . . . . . 6 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) → (¬ ∃!𝑥𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 → (𝐺 ∈ FriendGraph → ¬ 𝐺 ∈ FriendGraph )))
1312adantr 481 . . . . 5 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) ∧ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸))) → (¬ ∃!𝑥𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 → (𝐺 ∈ FriendGraph → ¬ 𝐺 ∈ FriendGraph )))
145, 13mpd 15 . . . 4 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) ∧ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸))) → (𝐺 ∈ FriendGraph → ¬ 𝐺 ∈ FriendGraph ))
1514pm2.01d 189 . . 3 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) ∧ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸))) → ¬ 𝐺 ∈ FriendGraph )
16 df-nel 3047 . . 3 (𝐺 ∉ FriendGraph ↔ ¬ 𝐺 ∈ FriendGraph )
1715, 16sylibr 233 . 2 (((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) ∧ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸))) → 𝐺 ∉ FriendGraph )
1817ex 413 1 ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) → ((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸)) → 𝐺 ∉ FriendGraph ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2940  wnel 3046  ∃!wreu 3374  wss 3947  {cpr 4629  cfv 6540  Vtxcvtx 28245  Edgcedg 28296  USGraphcusgr 28398   FriendGraph cfrgr 29500
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-frgr 29501
This theorem is referenced by:  frgrnbnb  29535  frgrwopreg  29565
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