Theorem wwlks2onv 26847

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 11309	. . . . . . . 8 ⊢ 2 ∈ ℕ0
2		wwlks2onv.v	. . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	wwlksnon 26738	. . . . . . . 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 706	. . . . . . 7 ⊢ (𝐺 ∈ V → (2 WWalksNOn 𝐺) = (𝑎 ∈ 𝑉, 𝑐 ∈ 𝑉 ↦ {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)}))
5	4	oveqd 6667	. . . . . 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 2687	. . . . 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 2622	. . . . . . 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 6876	. . . . . 6 ⊢ (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(𝑎 ∈ 𝑉, 𝑐 ∈ 𝑉 ↦ {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐)})𝐶) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
9		simprl 794	. . . . . . . 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 2633	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝐴 → ((𝑤‘0) = 𝑎 ↔ (𝑤‘0) = 𝐴))
11	10	anbi1d 741	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝐴 → (((𝑤‘0) = 𝑎 ∧ (𝑤‘2) = 𝑐) ↔ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝑐)))
12	11	rabbidv 3189	. . . . . . . . . . . . 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 2633	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝐶 → ((𝑤‘2) = 𝑐 ↔ (𝑤‘2) = 𝐶))
14	13	anbi2d 740	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝐶 → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝑐) ↔ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶)))
15	14	rabbidv 3189	. . . . . . . . . . . . 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 6678	. . . . . . . . . . . . . 14 ⊢ (2 WWalksN 𝐺) ∈ V
17	16	rabex 4813	. . . . . . . . . . . . 13 ⊢ {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶)} ∈ V
18	12, 15, 7, 17	ovmpt2 6796	. . . . . . . . . . . 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 2687	. . . . . . . . . . 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 6190	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ⟨“𝐴𝐵𝐶”⟩ → (𝑤‘0) = (⟨“𝐴𝐵𝐶”⟩‘0))
21	20	eqeq1d 2624	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ⟨“𝐴𝐵𝐶”⟩ → ((𝑤‘0) = 𝐴 ↔ (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴))
22		fveq1 6190	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ⟨“𝐴𝐵𝐶”⟩ → (𝑤‘2) = (⟨“𝐴𝐵𝐶”⟩‘2))
23	22	eqeq1d 2624	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ⟨“𝐴𝐵𝐶”⟩ → ((𝑤‘2) = 𝐶 ↔ (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶))
24	21, 23	anbi12d 747	. . . . . . . . . . . . 13 ⊢ (𝑤 = ⟨“𝐴𝐵𝐶”⟩ → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶) ↔ ((⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴 ∧ (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)))
25	24	elrab 3363	. . . . . . . . . . . 12 ⊢ (⟨“𝐴𝐵𝐶”⟩ ∈ {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶)} ↔ (⟨“𝐴𝐵𝐶”⟩ ∈ (2 WWalksN 𝐺) ∧ ((⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴 ∧ (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)))
26		wwlknbp2 26752	. . . . . . . . . . . . . . 15 ⊢ (⟨“𝐴𝐵𝐶”⟩ ∈ (2 WWalksN 𝐺) → (⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)))
27		s3fv1 13637	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ 𝑈 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
28	27	eqcomd 2628	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ 𝑈 → 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1))
29	28	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)) ∧ 𝐵 ∈ 𝑈) → 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1))
30		1ex 10035	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ V
31	30	tpid2 4304	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ {0, 1, 2}
32		id 22	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1) → (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1))
33		2p1e3 11151	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2 + 1) = 3
34	32, 33	syl6eq 2672	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1) → (#‘⟨“𝐴𝐵𝐶”⟩) = 3)
35	34	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1) → (0..^(#‘⟨“𝐴𝐵𝐶”⟩)) = (0..^3))
36		fzo0to3tp 12554	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0..^3) = {0, 1, 2}
37	35, 36	syl6eq 2672	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1) → (0..^(#‘⟨“𝐴𝐵𝐶”⟩)) = {0, 1, 2})
38	31, 37	syl5eleqr 2708	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1) → 1 ∈ (0..^(#‘⟨“𝐴𝐵𝐶”⟩)))
39		wrdsymbcl 13318	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ 1 ∈ (0..^(#‘⟨“𝐴𝐵𝐶”⟩))) → (⟨“𝐴𝐵𝐶”⟩‘1) ∈ (Vtx‘𝐺))
40	39, 2	syl6eleqr 2712	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ 1 ∈ (0..^(#‘⟨“𝐴𝐵𝐶”⟩))) → (⟨“𝐴𝐵𝐶”⟩‘1) ∈ 𝑉)
41	38, 40	sylan2 491	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)) → (⟨“𝐴𝐵𝐶”⟩‘1) ∈ 𝑉)
42	41	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)) ∧ 𝐵 ∈ 𝑈) → (⟨“𝐴𝐵𝐶”⟩‘1) ∈ 𝑉)
43	29, 42	eqeltrd 2701	. . . . . . . . . . . . . . . 16 ⊢ (((⟨“𝐴𝐵𝐶”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘⟨“𝐴𝐵𝐶”⟩) = (2 + 1)) ∧ 𝐵 ∈ 𝑈) → 𝐵 ∈ 𝑉)
44	43	ex 450	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 232	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (⟨“𝐴𝐵𝐶”⟩ ∈ {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐶)} → (𝐵 ∈ 𝑈 → 𝐵 ∈ 𝑉)))
49	19, 48	sylbid 230	. . . . . . . . . 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 447	. . . . . . . . 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 446	. . . . . . . 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 796	. . . . . . . 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 1242	. . . . . . 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 630	. . . . . 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 44	. . . . 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 243	. . . 4 ⊢ (𝐺 ∈ V → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐵 ∈ 𝑈 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))))
57	56	com23 86	. . 3 ⊢ (𝐺 ∈ V → (𝐵 ∈ 𝑈 → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))))
58	57	impd 447	. 2 ⊢ (𝐺 ∈ V → ((𝐵 ∈ 𝑈 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))
59		df-wwlksnon 26724	. . . . . . . . . 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 6771	. . . . . . . . 9 ⊢ Rel dom WWalksNOn
61	60	ovprc2 6685	. . . . . . . 8 ⊢ (¬ 𝐺 ∈ V → (2 WWalksNOn 𝐺) = ∅)
62	61	oveqd 6667	. . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (𝐴(2 WWalksNOn 𝐺)𝐶) = (𝐴∅𝐶))
63		0ov 6682	. . . . . . 7 ⊢ (𝐴∅𝐶) = ∅
64	62, 63	syl6eq 2672	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (𝐴(2 WWalksNOn 𝐺)𝐶) = ∅)
65	64	eleq2d 2687	. . . . 5 ⊢ (¬ 𝐺 ∈ V → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ⟨“𝐴𝐵𝐶”⟩ ∈ ∅))
66		noel 3919	. . . . . 6 ⊢ ¬ ⟨“𝐴𝐵𝐶”⟩ ∈ ∅
67	66	pm2.21i 116	. . . . 5 ⊢ (⟨“𝐴𝐵𝐶”⟩ ∈ ∅ → (𝐵 ∈ 𝑈 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))
68	65, 67	syl6bi 243	. . . 4 ⊢ (¬ 𝐺 ∈ V → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐵 ∈ 𝑈 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))))
69	68	com23 86	. . 3 ⊢ (¬ 𝐺 ∈ V → (𝐵 ∈ 𝑈 → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))))
70	69	impd 447	. 2 ⊢ (¬ 𝐺 ∈ V → ((𝐵 ∈ 𝑈 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))
71	58, 70	pm2.61i 176	1 ⊢ ((𝐵 ∈ 𝑈 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200  ∅c0 3915  {ctp 4181  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  0cc0 9936  1c1 9937   + caddc 9939  2c2 11070  3c3 11071  ℕ₀cn0 11292  ..^cfzo 12465  #chash 13117  Word cword 13291  ⟨“cs3 13587  Vtxcvtx 25874   WWalksN cwwlksn 26718   WWalksNOn cwwlksnon 26719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-wwlks 26722  df-wwlksn 26723  df-wwlksnon 26724

This theorem is referenced by:  frgr2wwlkeqm  27195