Theorem frgr2wwlk1 30361

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 30360	. . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (𝐴(2 WWalksNOn 𝐺)𝐵) ≠ ∅)
3	1	elwwlks2ons3 29988	. . . . . 6 ⊢ (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃𝑑 ∈ 𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
4	1	elwwlks2ons3 29988	. . . . . 6 ⊢ (𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃𝑐 ∈ 𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
5	3, 4	anbi12i 627	. . . . 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 30359	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → ∃!𝑥 ∈ 𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
7		s3eq2 14919	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ⟨“𝐴𝑥𝐵”⟩ = ⟨“𝐴𝑦𝐵”⟩)
8	7	eleq1d 2829	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
9	8	reu4 3753	. . . . . . . . . . . 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 14919	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑑 → ⟨“𝐴𝑥𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩)
11	10	eleq1d 2829	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑑 → (⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
12	11	anbi1d 630	. . . . . . . . . . . . . . . 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 2024	. . . . . . . . . . . . . . . 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 14919	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑐 → ⟨“𝐴𝑦𝐵”⟩ = ⟨“𝐴𝑐𝐵”⟩)
16	15	eleq1d 2829	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑐 → (⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
17	16	anbi2d 629	. . . . . . . . . . . . . . . 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 2025	. . . . . . . . . . . . . . . 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 3647	. . . . . . . . . . . . . 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 14919	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑐 = 𝑑 → ⟨“𝐴𝑐𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩)
23	22	equcoms 2019	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑑 = 𝑐 → ⟨“𝐴𝑐𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩)
24	23	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑑 = 𝑐 ∧ (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩)) → ⟨“𝐴𝑐𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩)
25		eqeq12 2757	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2018	. . . . . . . . . . . . . . . . . . . . . . . 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 3161	. . . . . . . . 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 3161	. . . . . . 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 1927	. . 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 4388	. . 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 583	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → ∃!𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
52		ovex 7481	. . 3 ⊢ (𝐴(2 WWalksNOn 𝐺)𝐵) ∈ V
53		euhash1 14469	. . 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 1087  ∀wal 1535   = wceq 1537   ∈ wcel 2108  ∃!weu 2571   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  ∃!wreu 3386  Vcvv 3488  ∅c0 4352  ‘cfv 6573  (class class class)co 7448  1c1 11185  2c2 12348  ♯chash 14379  ⟨“cs3 14891  Vtxcvtx 29031   WWalksNOn cwwlksnon 29860   FriendGraph cfrgr 30290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-ac2 10532  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-dju 9970  df-card 10008  df-ac 10185  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-xnn0 12626  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-hash 14380  df-word 14563  df-concat 14619  df-s1 14644  df-s2 14897  df-s3 14898  df-edg 29083  df-uhgr 29093  df-upgr 29117  df-umgr 29118  df-usgr 29186  df-wlks 29635  df-wwlks 29863  df-wwlksn 29864  df-wwlksnon 29865  df-frgr 30291

This theorem is referenced by:  frgr2wsp1  30362