Theorem wwlks2onv 41158

Description: If a length 3 string represents a walk of length 2, its components are vertices. (Contributed by Alexander van der Vekens, 19-Feb-2018.)

Hypothesis

Ref	Expression
wwlks2onv.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
wwlks2onv	⊢ ((𝐵 ∈ 𝑈 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))

Proof of Theorem wwlks2onv

Dummy variables 𝑎 𝑐 𝑤 𝑏 𝑔 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2nn0 11186	. . . . . . . 8 ⊢ 2 ∈ ℕ0
2		wwlks2onv.v	. . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	wwlksnon 41049	. . . . . . . 8 ⊢ ((2 ∈ ℕ0 ∧ 𝐺 ∈ V) → (2 WWalksNOn 𝐺) = (𝑎 ∈ 𝑉, 𝑐 ∈ 𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)}))
4	1, 3	mpan 702	. . . . . . 7 ⊢ (𝐺 ∈ V → (2 WWalksNOn 𝐺) = (𝑎 ∈ 𝑉, 𝑐 ∈ 𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)}))
5	4	oveqd 6566	. . . . . 6 ⊢ (𝐺 ∈ V → (𝐴(2 WWalksNOn 𝐺)𝐶) = (𝐴(𝑎 ∈ 𝑉, 𝑐 ∈ 𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶))
6	5	eleq2d 2673	. . . . 5 ⊢ (𝐺 ∈ V → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎 ∈ 𝑉, 𝑐 ∈ 𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶)))
7		eqid 2610	. . . . . . 7 ⊢ (𝑎 ∈ 𝑉, 𝑐 ∈ 𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)}) = (𝑎 ∈ 𝑉, 𝑐 ∈ 𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})
8	7	elmpt2cl 6774	. . . . . 6 ⊢ (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎 ∈ 𝑉, 𝑐 ∈ 𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
9		simprl 790	. . . . . . . 8 ⊢ (((⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎 ∈ 𝑉, 𝑐 ∈ 𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) ∧ 𝐵 ∈ 𝑈) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐴 ∈ 𝑉)
10		eqeq2 2621	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝐴 → ((𝑤‘0) = 𝑎 ↔ (𝑤‘0) = 𝐴))
11	10	anbi1d 737	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝐴 → (((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐) ↔ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝑐)))
12	11	rabbidv 3164	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)} = {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝑐)})
13		eqeq2 2621	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝐶 → ((𝑤‘2) = 𝑐 ↔ (𝑤‘2) = 𝐶))
14	13	anbi2d 736	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝐶 → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝑐) ↔ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶)))
15	14	rabbidv 3164	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝐶 → {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝑐)} = {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶)})
16		ovex 6577	. . . . . . . . . . . . . 14 ⊢ (2 WWalkSN 𝐺) ∈ V
17	16	rabex 4740	. . . . . . . . . . . . 13 ⊢ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶)} ∈ V
18	12, 15, 7, 17	ovmpt2 6694	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴(𝑎 ∈ 𝑉, 𝑐 ∈ 𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) = {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶)})
19	18	eleq2d 2673	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎 ∈ 𝑉, 𝑐 ∈ 𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) ↔ ⟨“𝐴𝐵𝐶”⟩ ∈ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶)}))
20		fveq1 6102	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ⟨“𝐴𝐵𝐶”⟩ → (𝑤‘0) = (⟨“𝐴𝐵𝐶”⟩‘0))
21	20	eqeq1d 2612	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ⟨“𝐴𝐵𝐶”⟩ → ((𝑤‘0) = 𝐴 ↔ (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴))
22		fveq1 6102	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ⟨“𝐴𝐵𝐶”⟩ → (𝑤‘2) = (⟨“𝐴𝐵𝐶”⟩‘2))
23	22	eqeq1d 2612	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ⟨“𝐴𝐵𝐶”⟩ → ((𝑤‘2) = 𝐶 ↔ (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶))
24	21, 23	anbi12d 743	. . . . . . . . . . . . 13 ⊢ (𝑤 = ⟨“𝐴𝐵𝐶”⟩ → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶) ↔ ((⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴 ∧ (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)))
25	24	elrab 3331	. . . . . . . . . . . 12 ⊢ (⟨“𝐴𝐵𝐶”⟩ ∈ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶)} ↔ (⟨“𝐴𝐵𝐶”⟩ ∈ (2 WWalkSN 𝐺) ∧ ((⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴 ∧ (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)))
26		wwlknbp2 41063	. . . . . . . . . . . . . . 15 ⊢ (⟨“𝐴𝐵𝐶”⟩ ∈ (2 WWalkSN 𝐺) → (⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)))
27		s3fv1 13487	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ 𝑈 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
28	27	eqcomd 2616	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ 𝑈 → 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1))
29	28	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)) ∧ 𝐵 ∈ 𝑈) → 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1))
30		1ex 9914	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ V
31	30	tpid2 4247	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ {0, 1, 2}
32		id 22	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1) → (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1))
33		2p1e3 11028	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2 + 1) = 3
34	32, 33	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1) → (#‘⟨“𝐴𝐵𝐶”⟩) = 3)
35	34	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1) → (0..^(#‘⟨“𝐴𝐵𝐶”⟩)) = (0..^3))
36		fzo0to3tp 12421	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0..^3) = {0, 1, 2}
37	35, 36	syl6eq 2660	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1) → (0..^(#‘⟨“𝐴𝐵𝐶”⟩)) = {0, 1, 2})
38	31, 37	syl5eleqr 2695	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1) → 1 ∈ (0..^(#‘⟨“𝐴𝐵𝐶”⟩)))
39		wrdsymbcl 13173	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ 1 ∈ (0..^(#‘⟨“𝐴𝐵𝐶”⟩))) → (⟨“𝐴𝐵𝐶”⟩‘1) ∈ (Vtx‘𝐺))
40	39, 2	syl6eleqr 2699	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ 1 ∈ (0..^(#‘⟨“𝐴𝐵𝐶”⟩))) → (⟨“𝐴𝐵𝐶”⟩‘1) ∈ 𝑉)
41	38, 40	sylan2 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)) → (⟨“𝐴𝐵𝐶”⟩‘1) ∈ 𝑉)
42	41	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)) ∧ 𝐵 ∈ 𝑈) → (⟨“𝐴𝐵𝐶”⟩‘1) ∈ 𝑉)
43	29, 42	eqeltrd 2688	. . . . . . . . . . . . . . . 16 ⊢ (((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)) ∧ 𝐵 ∈ 𝑈) → 𝐵 ∈ 𝑉)
44	43	ex 449	. . . . . . . . . . . . . . 15 ⊢ ((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)) → (𝐵 ∈ 𝑈 → 𝐵 ∈ 𝑉))
45	26, 44	syl 17	. . . . . . . . . . . . . 14 ⊢ (⟨“𝐴𝐵𝐶”⟩ ∈ (2 WWalkSN 𝐺) → (𝐵 ∈ 𝑈 → 𝐵 ∈ 𝑉))
46	45	adantr 480	. . . . . . . . . . . . 13 ⊢ ((⟨“𝐴𝐵𝐶”⟩ ∈ (2 WWalkSN 𝐺) ∧ ((⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴 ∧ (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)) → (𝐵 ∈ 𝑈 → 𝐵 ∈ 𝑉))
47	46	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((⟨“𝐴𝐵𝐶”⟩ ∈ (2 WWalkSN 𝐺) ∧ ((⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴 ∧ (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)) → (𝐵 ∈ 𝑈 → 𝐵 ∈ 𝑉)))
48	25, 47	syl5bi 231	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (⟨“𝐴𝐵𝐶”⟩ ∈ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶)} → (𝐵 ∈ 𝑈 → 𝐵 ∈ 𝑉)))
49	19, 48	sylbid 229	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎 ∈ 𝑉, 𝑐 ∈ 𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) → (𝐵 ∈ 𝑈 → 𝐵 ∈ 𝑉)))
50	49	impd 446	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎 ∈ 𝑉, 𝑐 ∈ 𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) ∧ 𝐵 ∈ 𝑈) → 𝐵 ∈ 𝑉))
51	50	impcom 445	. . . . . . . 8 ⊢ (((⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎 ∈ 𝑉, 𝑐 ∈ 𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) ∧ 𝐵 ∈ 𝑈) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐵 ∈ 𝑉)
52		simprr 792	. . . . . . . 8 ⊢ (((⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎 ∈ 𝑉, 𝑐 ∈ 𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) ∧ 𝐵 ∈ 𝑈) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ 𝑉)
53	9, 51, 52	3jca 1235	. . . . . . 7 ⊢ (((⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎 ∈ 𝑉, 𝑐 ∈ 𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) ∧ 𝐵 ∈ 𝑈) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
54	53	exp31 628	. . . . . 6 ⊢ (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎 ∈ 𝑉, 𝑐 ∈ 𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) → (𝐵 ∈ 𝑈 → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))))
55	8, 54	mpid 43	. . . . 5 ⊢ (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎 ∈ 𝑉, 𝑐 ∈ 𝑉 ↦ {𝑤 ∈ (2 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) → (𝐵 ∈ 𝑈 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))
56	6, 55	syl6bi 242	. . . 4 ⊢ (𝐺 ∈ V → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐵 ∈ 𝑈 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))))
57	56	com23 84	. . 3 ⊢ (𝐺 ∈ V → (𝐵 ∈ 𝑈 → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))))
58	57	impd 446	. 2 ⊢ (𝐺 ∈ V → ((𝐵 ∈ 𝑈 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))
59		df-wwlksnon 41035	. . . . . . . . . 10 ⊢ WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalkSN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑛) = 𝑏)}))
60	59	reldmmpt2 6669	. . . . . . . . 9 ⊢ Rel dom WWalksNOn
61	60	ovprc2 6583	. . . . . . . 8 ⊢ (¬ 𝐺 ∈ V → (2 WWalksNOn 𝐺) = ∅)
62	61	oveqd 6566	. . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (𝐴(2 WWalksNOn 𝐺)𝐶) = (𝐴∅𝐶))
63		0ov 6580	. . . . . . 7 ⊢ (𝐴∅𝐶) = ∅
64	62, 63	syl6eq 2660	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (𝐴(2 WWalksNOn 𝐺)𝐶) = ∅)
65	64	eleq2d 2673	. . . . 5 ⊢ (¬ 𝐺 ∈ V → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ⟨“𝐴𝐵𝐶”⟩ ∈ ∅))
66		noel 3878	. . . . . 6 ⊢ ¬ ⟨“𝐴𝐵𝐶”⟩ ∈ ∅
67	66	pm2.21i 115	. . . . 5 ⊢ (⟨“𝐴𝐵𝐶”⟩ ∈ ∅ → (𝐵 ∈ 𝑈 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))
68	65, 67	syl6bi 242	. . . 4 ⊢ (¬ 𝐺 ∈ V → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐵 ∈ 𝑈 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))))
69	68	com23 84	. . 3 ⊢ (¬ 𝐺 ∈ V → (𝐵 ∈ 𝑈 → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))))
70	69	impd 446	. 2 ⊢ (¬ 𝐺 ∈ V → ((𝐵 ∈ 𝑈 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))
71	58, 70	pm2.61i 175	1 ⊢ ((𝐵 ∈ 𝑈 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  ∅c0 3874  {ctp 4129  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  1c1 9816   + caddc 9818  2c2 10947  3c3 10948  ℕ₀cn0 11169  ..^cfzo 12334  #chash 12979  Word cword 13146  ⟨“cs3 13438  Vtxcvtx 25673   WWalkSN cwwlksn 41029   WWalksNOn cwwlksnon 41030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-s2 13444  df-s3 13445  df-wwlks 41033  df-wwlksn 41034  df-wwlksnon 41035

This theorem is referenced by:  frgr2wwlkeqm  41496