Theorem frgr2wwlkeqm 30267

Description: If there is a (simple) path of length 2 from one vertex to another vertex and a (simple) path of length 2 from the other vertex back to the first vertex in a friendship graph, then the middle vertex is the same. This is only an observation, which is not required to proof the friendship theorem. (Contributed by Alexander van der Vekens, 20-Feb-2018.) (Revised by AV, 13-May-2021.) (Proof shortened by AV, 7-Jan-2022.)

Assertion

Ref	Expression
frgr2wwlkeqm	⊢ ((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → ((⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → 𝑄 = 𝑃))

Proof of Theorem frgr2wwlkeqm

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3l 1202	. . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → 𝑃 ∈ 𝑋)
2		eqid 2730	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3	2	wwlks2onv 29890	. . . 4 ⊢ ((𝑃 ∈ 𝑋 ∧ ⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
4	1, 3	sylan 580	. . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
5		simp3r 1203	. . . . . . . 8 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → 𝑄 ∈ 𝑌)
6	2	wwlks2onv 29890	. . . . . . . 8 ⊢ ((𝑄 ∈ 𝑌 ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)))
7	5, 6	sylan 580	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)))
8		frgrusgr 30197	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
9		usgrumgr 29115	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
10	8, 9	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ UMGraph)
11	10	3ad2ant1 1133	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → 𝐺 ∈ UMGraph)
12		simpr3 1197	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → 𝐵 ∈ (Vtx‘𝐺))
13		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → 𝑄 ∈ (Vtx‘𝐺))
14		simpr1 1195	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → 𝐴 ∈ (Vtx‘𝐺))
15	12, 13, 14	3jca 1128	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)))
16	2	wwlks2onsym 29895	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ UMGraph ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) ↔ ⟨“𝐴𝑄𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
17	11, 15, 16	syl2anr 597	. . . . . . . . . . . 12 ⊢ (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) ↔ ⟨“𝐴𝑄𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
18		simpr1 1195	. . . . . . . . . . . . . 14 ⊢ (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → 𝐺 ∈ FriendGraph )
19		3simpb 1149	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
20	19	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
21		simpr2 1196	. . . . . . . . . . . . . 14 ⊢ (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → 𝐴 ≠ 𝐵)
22	2	frgr2wwlkeu 30263	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ 𝐴 ≠ 𝐵) → ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
23	18, 20, 21, 22	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
24		s3eq2 14843	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑄 → ⟨“𝐴𝑥𝐵”⟩ = ⟨“𝐴𝑄𝐵”⟩)
25	24	eleq1d 2814	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑄 → (⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑄𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
26	25	riota2 7372	. . . . . . . . . . . . . . 15 ⊢ ((𝑄 ∈ (Vtx‘𝐺) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐴𝑄𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (℩𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑄))
27	26	ad4ant14 752	. . . . . . . . . . . . . 14 ⊢ ((((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐴𝑄𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (℩𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑄))
28		simplr2 1217	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → 𝑃 ∈ (Vtx‘𝐺))
29		s3eq2 14843	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑃 → ⟨“𝐴𝑥𝐵”⟩ = ⟨“𝐴𝑃𝐵”⟩)
30	29	eleq1d 2814	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑃 → (⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
31	30	riota2 7372	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ (Vtx‘𝐺) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (℩𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑃))
32	28, 31	sylan 580	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (℩𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑃))
33		eqtr2 2751	. . . . . . . . . . . . . . . . 17 ⊢ (((℩𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑄 ∧ (℩𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑃) → 𝑄 = 𝑃)
34	33	expcom 413	. . . . . . . . . . . . . . . 16 ⊢ ((℩𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑃 → ((℩𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑄 → 𝑄 = 𝑃))
35	32, 34	biimtrdi 253	. . . . . . . . . . . . . . 15 ⊢ ((((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → ((℩𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑄 → 𝑄 = 𝑃)))
36	35	com23 86	. . . . . . . . . . . . . 14 ⊢ ((((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((℩𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) = 𝑄 → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
37	27, 36	sylbid 240	. . . . . . . . . . . . 13 ⊢ ((((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) ∧ ∃!𝑥 ∈ (Vtx‘𝐺)⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐴𝑄𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
38	23, 37	mpdan 687	. . . . . . . . . . . 12 ⊢ (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → (⟨“𝐴𝑄𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
39	17, 38	sylbid 240	. . . . . . . . . . 11 ⊢ (((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌))) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
40	39	expimpd 453	. . . . . . . . . 10 ⊢ ((𝑄 ∈ (Vtx‘𝐺) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → (((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
41	40	ex 412	. . . . . . . . 9 ⊢ (𝑄 ∈ (Vtx‘𝐺) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃))))
42	41	com23 86	. . . . . . . 8 ⊢ (𝑄 ∈ (Vtx‘𝐺) → (((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃))))
43	42	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) → (((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃))))
44	7, 43	mpcom 38	. . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃)))
45	44	ex 412	. . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → 𝑄 = 𝑃))))
46	45	com24 95	. . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) → 𝑄 = 𝑃))))
47	46	imp 406	. . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) → 𝑄 = 𝑃)))
48	4, 47	mpd 15	. 2 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) ∧ ⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) → 𝑄 = 𝑃))
49	48	expimpd 453	1 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → ((⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → 𝑄 = 𝑃))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∃!wreu 3354  ‘cfv 6514  ℩crio 7346  (class class class)co 7390  2c2 12248  ⟨“cs3 14815  Vtxcvtx 28930  UMGraphcumgr 29015  USGraphcusgr 29083   WWalksNOn cwwlksnon 29764   FriendGraph cfrgr 30194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-ac2 10423  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-oadd 8441  df-er 8674  df-map 8804  df-pm 8805  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-dju 9861  df-card 9899  df-ac 10076  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-xnn0 12523  df-z 12537  df-uz 12801  df-fz 13476  df-fzo 13623  df-hash 14303  df-word 14486  df-concat 14543  df-s1 14568  df-s2 14821  df-s3 14822  df-edg 28982  df-uhgr 28992  df-upgr 29016  df-umgr 29017  df-usgr 29085  df-wlks 29534  df-wwlks 29767  df-wwlksn 29768  df-wwlksnon 29769  df-frgr 30195

This theorem is referenced by: (None)