Theorem frgr2wwlk1 30308

Description: In a friendship graph, there is exactly one walk of length 2 between two different vertices. (Contributed by Alexander van der Vekens, 19-Feb-2018.) (Revised by AV, 13-May-2021.) (Proof shortened by AV, 16-Mar-2022.)

Hypothesis

Ref	Expression
frgr2wwlkeu.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
frgr2wwlk1	⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (♯‘(𝐴(2 WWalksNOn 𝐺)𝐵)) = 1)

Proof of Theorem frgr2wwlk1

Dummy variables 𝑐 𝑑 𝑡 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgr2wwlkeu.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	frgr2wwlkn0 30307	. . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (𝐴(2 WWalksNOn 𝐺)𝐵) ≠ ∅)
3	1	elwwlks2ons3 29935	. . . . . 6 ⊢ (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃𝑑 ∈ 𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
4	1	elwwlks2ons3 29935	. . . . . 6 ⊢ (𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃𝑐 ∈ 𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
5	3, 4	anbi12i 628	. . . . 5 ⊢ ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ↔ (∃𝑑 ∈ 𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ∧ ∃𝑐 ∈ 𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))))
6	1	frgr2wwlkeu 30306	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → ∃!𝑥 ∈ 𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
7		s3eq2 14812	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ⟨“𝐴𝑥𝐵”⟩ = ⟨“𝐴𝑦𝐵”⟩)
8	7	eleq1d 2813	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
9	8	reu4 3699	. . . . . . . . . . . 12 ⊢ (∃!𝑥 ∈ 𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (∃𝑥 ∈ 𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ((⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑥 = 𝑦)))
10		s3eq2 14812	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑑 → ⟨“𝐴𝑥𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩)
11	10	eleq1d 2813	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑑 → (⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
12	11	anbi1d 631	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑑 → ((⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ↔ (⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))))
13		equequ1 2025	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑑 → (𝑥 = 𝑦 ↔ 𝑑 = 𝑦))
14	12, 13	imbi12d 344	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑑 → (((⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑥 = 𝑦) ↔ ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑦)))
15		s3eq2 14812	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑐 → ⟨“𝐴𝑦𝐵”⟩ = ⟨“𝐴𝑐𝐵”⟩)
16	15	eleq1d 2813	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑐 → (⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
17	16	anbi2d 630	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑐 → ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ↔ (⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))))
18		equequ2 2026	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑐 → (𝑑 = 𝑦 ↔ 𝑑 = 𝑐))
19	17, 18	imbi12d 344	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑐 → (((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑦) ↔ ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐)))
20	14, 19	rspc2va 3597	. . . . . . . . . . . . . 14 ⊢ (((𝑑 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ((⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑥 = 𝑦)) → ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐))
21		pm3.35 802	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ∧ ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐)) → 𝑑 = 𝑐)
22		s3eq2 14812	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑐 = 𝑑 → ⟨“𝐴𝑐𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩)
23	22	equcoms 2020	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑑 = 𝑐 → ⟨“𝐴𝑐𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩)
24	23	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑑 = 𝑐 ∧ (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩)) → ⟨“𝐴𝑐𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩)
25		eqeq12 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩) → (𝑡 = 𝑤 ↔ ⟨“𝐴𝑐𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩))
26	25	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑑 = 𝑐 ∧ (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩)) → (𝑡 = 𝑤 ↔ ⟨“𝐴𝑐𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩))
27	24, 26	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑑 = 𝑐 ∧ (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩)) → 𝑡 = 𝑤)
28	27	equcomd 2019	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑑 = 𝑐 ∧ (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩)) → 𝑤 = 𝑡)
29	28	ex 412	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑑 = 𝑐 → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩) → 𝑤 = 𝑡))
30	21, 29	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ∧ ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐)) → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩) → 𝑤 = 𝑡))
31	30	ex 412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐) → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩) → 𝑤 = 𝑡)))
32	31	com23 86	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩) → (((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐) → 𝑤 = 𝑡)))
33	32	exp4b 430	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → (⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → (𝑡 = ⟨“𝐴𝑐𝐵”⟩ → (𝑤 = ⟨“𝐴𝑑𝐵”⟩ → (((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐) → 𝑤 = 𝑡)))))
34	33	com13 88	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = ⟨“𝐴𝑐𝐵”⟩ → (⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → (⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → (𝑤 = ⟨“𝐴𝑑𝐵”⟩ → (((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐) → 𝑤 = 𝑡)))))
35	34	imp 406	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → (𝑤 = ⟨“𝐴𝑑𝐵”⟩ → (((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐) → 𝑤 = 𝑡))))
36	35	com13 88	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = ⟨“𝐴𝑑𝐵”⟩ → (⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐) → 𝑤 = 𝑡))))
37	36	imp 406	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐) → 𝑤 = 𝑡)))
38	37	com13 88	. . . . . . . . . . . . . 14 ⊢ (((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐) → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡)))
39	20, 38	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑑 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ((⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑥 = 𝑦)) → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡)))
40	39	expcom 413	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ((⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑥 = 𝑦) → ((𝑑 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))))
41	9, 40	simplbiim 504	. . . . . . . . . . 11 ⊢ (∃!𝑥 ∈ 𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → ((𝑑 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))))
42	41	impl 455	. . . . . . . . . 10 ⊢ (((∃!𝑥 ∈ 𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 ∈ 𝑉) → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡)))
43	42	rexlimdva 3134	. . . . . . . . 9 ⊢ ((∃!𝑥 ∈ 𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑑 ∈ 𝑉) → (∃𝑐 ∈ 𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡)))
44	43	com23 86	. . . . . . . 8 ⊢ ((∃!𝑥 ∈ 𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑑 ∈ 𝑉) → ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (∃𝑐 ∈ 𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡)))
45	44	rexlimdva 3134	. . . . . . 7 ⊢ (∃!𝑥 ∈ 𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → (∃𝑑 ∈ 𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (∃𝑐 ∈ 𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡)))
46	45	impd 410	. . . . . 6 ⊢ (∃!𝑥 ∈ 𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → ((∃𝑑 ∈ 𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ∧ ∃𝑐 ∈ 𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))) → 𝑤 = 𝑡))
47	6, 46	syl 17	. . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → ((∃𝑑 ∈ 𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ∧ ∃𝑐 ∈ 𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))) → 𝑤 = 𝑡))
48	5, 47	biimtrid 242	. . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))
49	48	alrimivv 1928	. . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → ∀𝑤∀𝑡((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))
50		eqeuel 4324	. . 3 ⊢ (((𝐴(2 WWalksNOn 𝐺)𝐵) ≠ ∅ ∧ ∀𝑤∀𝑡((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡)) → ∃!𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
51	2, 49, 50	syl2anc 584	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → ∃!𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
52		ovex 7402	. . 3 ⊢ (𝐴(2 WWalksNOn 𝐺)𝐵) ∈ V
53		euhash1 14361	. . 3 ⊢ ((𝐴(2 WWalksNOn 𝐺)𝐵) ∈ V → ((♯‘(𝐴(2 WWalksNOn 𝐺)𝐵)) = 1 ↔ ∃!𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
54	52, 53	mp1i 13	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → ((♯‘(𝐴(2 WWalksNOn 𝐺)𝐵)) = 1 ↔ ∃!𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
55	51, 54	mpbird 257	1 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (♯‘(𝐴(2 WWalksNOn 𝐺)𝐵)) = 1)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∃!weu 2561   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  ∃!wreu 3349  Vcvv 3444  ∅c0 4292  ‘cfv 6499  (class class class)co 7369  1c1 11045  2c2 12217  ♯chash 14271  ⟨“cs3 14784  Vtxcvtx 28976   WWalksNOn cwwlksnon 29807   FriendGraph cfrgr 30237

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-ac2 10392  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-oadd 8415  df-er 8648  df-map 8778  df-pm 8779  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-dju 9830  df-card 9868  df-ac 10045  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-n0 12419  df-xnn0 12492  df-z 12506  df-uz 12770  df-fz 13445  df-fzo 13592  df-hash 14272  df-word 14455  df-concat 14512  df-s1 14537  df-s2 14790  df-s3 14791  df-edg 29028  df-uhgr 29038  df-upgr 29062  df-umgr 29063  df-usgr 29131  df-wlks 29580  df-wwlks 29810  df-wwlksn 29811  df-wwlksnon 29812  df-frgr 30238

This theorem is referenced by:  frgr2wsp1  30309