Theorem 4cyclusnfrgr 30496

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 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷)) → ((({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.

Step	Hyp	Ref	Expression
1		simprl 780	. . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷)) ∧ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸))) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))
2		simprr 782	. . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷)) ∧ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸))) → ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸))
3		simpl3 1208	. . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷)) ∧ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸))) → (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷))
4		4cycl2vnunb 30494	. . . . . 6 ⊢ ((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷)) → ¬ ∃!𝑥 ∈ 𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸)
5	1, 2, 3, 4	syl3anc 1392	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷)) ∧ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸))) → ¬ ∃!𝑥 ∈ 𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸)
6		4cyclusnfrgr.v	. . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺)
7		4cyclusnfrgr.e	. . . . . . . . . 10 ⊢ 𝐸 = (Edg‘𝐺)
8	6, 7	frcond1 30470	. . . . . . . . 9 ⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑥 ∈ 𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸))
9		pm2.24 124	. . . . . . . . 9 ⊢ (∃!𝑥 ∈ 𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 → (¬ ∃!𝑥 ∈ 𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 → ¬ 𝐺 ∈ FriendGraph ))
10	8, 9	syl6com 37	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → (𝐺 ∈ FriendGraph → (¬ ∃!𝑥 ∈ 𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 → ¬ 𝐺 ∈ FriendGraph )))
11	10	3ad2ant2 1148	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷)) → (𝐺 ∈ FriendGraph → (¬ ∃!𝑥 ∈ 𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 → ¬ 𝐺 ∈ FriendGraph )))
12	11	com23 86	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷)) → (¬ ∃!𝑥 ∈ 𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 → (𝐺 ∈ FriendGraph → ¬ 𝐺 ∈ FriendGraph )))
13	12	adantr 484	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷)) ∧ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸))) → (¬ ∃!𝑥 ∈ 𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 → (𝐺 ∈ FriendGraph → ¬ 𝐺 ∈ FriendGraph )))
14	5, 13	mpd 15	. . . 4 ⊢ (((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷)) ∧ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸))) → (𝐺 ∈ FriendGraph → ¬ 𝐺 ∈ FriendGraph ))
15	14	pm2.01d 191	. . 3 ⊢ (((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷)) ∧ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸))) → ¬ 𝐺 ∈ FriendGraph )
16		df-nel 3064	. . 3 ⊢ (𝐺 ∉ FriendGraph ↔ ¬ 𝐺 ∈ FriendGraph )
17	15, 16	sylibr 236	. 2 ⊢ (((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷)) ∧ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸))) → 𝐺 ∉ FriendGraph )
18	17	ex 416	1 ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷)) → ((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸)) → 𝐺 ∉ FriendGraph ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144   ≠ wne 2959   ∉ wnel 3063  ∃!wreu 3367   ⊆ wss 3906  {cpr 4586  ‘cfv 6523  Vtxcvtx 29199  Edgcedg 29250  USGraphcusgr 29352   FriendGraph cfrgr 30462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-nul 5258

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-iota 6479  df-fv 6531  df-frgr 30463

This theorem is referenced by:  frgrnbnb  30497  frgrwopreg  30527