Theorem 2pthon3v 30537

Description: For a vertex adjacent to two other vertices there is a simple path of length 2 between these other vertices in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 24-Jan-2021.)

Hypotheses

Ref	Expression
2pthon3v.v	⊢ 𝑉 = (Vtx‘𝐺)
2pthon3v.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
2pthon3v	⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)) → ∃𝑓∃𝑝(𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (♯‘𝑓) = 2))

Distinct variable groups:   𝐴,𝑓,𝑝   𝐵,𝑓,𝑝   𝐶,𝑓,𝑝   𝑓,𝐺,𝑝

Allowed substitution hints:   𝐸(𝑓, 𝑝)   𝑉(𝑓, 𝑝)

Proof of Theorem 2pthon3v

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2pthon3v.e	. . . . . . . . . 10 ⊢ 𝐸 = (Edg‘𝐺)
2		edgval 29632	. . . . . . . . . 10 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
3	1, 2	eqtri 2783	. . . . . . . . 9 ⊢ 𝐸 = ran (iEdg‘𝐺)
4	3	eleq2i 2852	. . . . . . . 8 ⊢ ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ ran (iEdg‘𝐺))
5		2pthon3v.v	. . . . . . . . . . 11 ⊢ 𝑉 = (Vtx‘𝐺)
6		eqid 2760	. . . . . . . . . . 11 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
7	5, 6	uhgrf 29645	. . . . . . . . . 10 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
8	7	ffnd 6710	. . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
9		fvelrnb 6945	. . . . . . . . 9 ⊢ ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → ({𝐴, 𝐵} ∈ ran (iEdg‘𝐺) ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵}))
10	8, 9	syl 18	. . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → ({𝐴, 𝐵} ∈ ran (iEdg‘𝐺) ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵}))
11	4, 10	bitrid 286	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → ({𝐴, 𝐵} ∈ 𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵}))
12	3	eleq2i 2852	. . . . . . . 8 ⊢ ({𝐵, 𝐶} ∈ 𝐸 ↔ {𝐵, 𝐶} ∈ ran (iEdg‘𝐺))
13		fvelrnb 6945	. . . . . . . . 9 ⊢ ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → ({𝐵, 𝐶} ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}))
14	8, 13	syl 18	. . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → ({𝐵, 𝐶} ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}))
15	12, 14	bitrid 286	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → ({𝐵, 𝐶} ∈ 𝐸 ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}))
16	11, 15	anbi12d 644	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ↔ (∃𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})))
17	16	adantr 486	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ↔ (∃𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})))
18	17	adantr 486	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ↔ (∃𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})))
19		reeanv 3234	. . . 4 ⊢ (∃𝑖 ∈ dom (iEdg‘𝐺)∃𝑗 ∈ dom (iEdg‘𝐺)(((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) ↔ (∃𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}))
20	18, 19	bitr4di 292	. . 3 ⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)∃𝑗 ∈ dom (iEdg‘𝐺)(((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})))
21		df-s2 15003	. . . . . . . 8 ⊢ ⟨“𝑖𝑗”⟩ = (⟨“𝑖”⟩ ++ ⟨“𝑗”⟩)
22	21	ovexi 7454	. . . . . . 7 ⊢ ⟨“𝑖𝑗”⟩ ∈ V
23		df-s3 15004	. . . . . . . 8 ⊢ ⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩)
24	23	ovexi 7454	. . . . . . 7 ⊢ ⟨“𝐴𝐵𝐶”⟩ ∈ V
25	22, 24	pm3.2i 476	. . . . . 6 ⊢ (⟨“𝑖𝑗”⟩ ∈ V ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ V)
26		eqid 2760	. . . . . . . 8 ⊢ ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐵𝐶”⟩
27		eqid 2760	. . . . . . . 8 ⊢ ⟨“𝑖𝑗”⟩ = ⟨“𝑖𝑗”⟩
28		simp-4r 796	. . . . . . . 8 ⊢ (((((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) ∧ (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
29		3simpb 1167	. . . . . . . . 9 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶))
30	29	ad3antlr 744	. . . . . . . 8 ⊢ (((((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) ∧ (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})) → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶))
31		eqimss2 3967	. . . . . . . . . 10 ⊢ (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
32		eqimss2 3967	. . . . . . . . . 10 ⊢ (((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶} → {𝐵, 𝐶} ⊆ ((iEdg‘𝐺)‘𝑗))
33	31, 32	anim12i 625	. . . . . . . . 9 ⊢ ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖) ∧ {𝐵, 𝐶} ⊆ ((iEdg‘𝐺)‘𝑗)))
34	33	adantl 487	. . . . . . . 8 ⊢ (((((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) ∧ (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})) → ({𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖) ∧ {𝐵, 𝐶} ⊆ ((iEdg‘𝐺)‘𝑗)))
35		fveqeq2 6894	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ↔ ((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵}))
36	35	anbi1d 643	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) ↔ (((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})))
37		eqtr2 2781	. . . . . . . . . . . . . 14 ⊢ ((((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → {𝐴, 𝐵} = {𝐵, 𝐶})
38		3simpa 1166	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))
39		3simpc 1168	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
40		preq12bg 4807	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐵))))
41	38, 39, 40	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐵))))
42		eqneqall 2966	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝑖 ≠ 𝑗))
43	42	com12 33	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 ≠ 𝐵 → (𝐴 = 𝐵 → 𝑖 ≠ 𝑗))
44	43	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → (𝐴 = 𝐵 → 𝑖 ≠ 𝑗))
45	44	com12 33	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 = 𝐵 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝑖 ≠ 𝑗))
46	45	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝑖 ≠ 𝑗))
47		eqneqall 2966	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 = 𝐶 → (𝐴 ≠ 𝐶 → 𝑖 ≠ 𝑗))
48	47	com12 33	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 ≠ 𝐶 → (𝐴 = 𝐶 → 𝑖 ≠ 𝑗))
49	48	3ad2ant2 1152	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → (𝐴 = 𝐶 → 𝑖 ≠ 𝑗))
50	49	com12 33	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 = 𝐶 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝑖 ≠ 𝑗))
51	50	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐵) → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝑖 ≠ 𝑗))
52	46, 51	jaoi 871	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐵)) → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝑖 ≠ 𝑗))
53	41, 52	biimtrdi 256	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ({𝐴, 𝐵} = {𝐵, 𝐶} → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝑖 ≠ 𝑗)))
54	53	com23 87	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 𝑖 ≠ 𝑗)))
55	54	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 𝑖 ≠ 𝑗)))
56	55	imp 412	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 𝑖 ≠ 𝑗))
57	56	com12 33	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐵} = {𝐵, 𝐶} → (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → 𝑖 ≠ 𝑗))
58	37, 57	syl 18	. . . . . . . . . . . . 13 ⊢ ((((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → 𝑖 ≠ 𝑗))
59	36, 58	biimtrdi 256	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → 𝑖 ≠ 𝑗)))
60	59	com23 87	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → 𝑖 ≠ 𝑗)))
61		2a1 29	. . . . . . . . . . 11 ⊢ (𝑖 ≠ 𝑗 → (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → 𝑖 ≠ 𝑗)))
62	60, 61	pm2.61ine 3038	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → 𝑖 ≠ 𝑗))
63	62	adantr 486	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) → ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → 𝑖 ≠ 𝑗))
64	63	imp 412	. . . . . . . 8 ⊢ (((((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) ∧ (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})) → 𝑖 ≠ 𝑗)
65		simplr2 1235	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) → 𝐴 ≠ 𝐶)
66	65	adantr 486	. . . . . . . 8 ⊢ (((((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) ∧ (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})) → 𝐴 ≠ 𝐶)
67	26, 27, 28, 30, 34, 5, 6, 64, 66	2pthond 30536	. . . . . . 7 ⊢ (((((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) ∧ (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})) → ⟨“𝑖𝑗”⟩(𝐴(SPathsOn‘𝐺)𝐶)⟨“𝐴𝐵𝐶”⟩)
68		s2len 15044	. . . . . . 7 ⊢ (♯‘⟨“𝑖𝑗”⟩) = 2
69	67, 68	jctir 530	. . . . . 6 ⊢ (((((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) ∧ (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})) → (⟨“𝑖𝑗”⟩(𝐴(SPathsOn‘𝐺)𝐶)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘⟨“𝑖𝑗”⟩) = 2))
70		breq12 5107	. . . . . . . 8 ⊢ ((𝑓 = ⟨“𝑖𝑗”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → (𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ↔ ⟨“𝑖𝑗”⟩(𝐴(SPathsOn‘𝐺)𝐶)⟨“𝐴𝐵𝐶”⟩))
71		fveqeq2 6894	. . . . . . . . 9 ⊢ (𝑓 = ⟨“𝑖𝑗”⟩ → ((♯‘𝑓) = 2 ↔ (♯‘⟨“𝑖𝑗”⟩) = 2))
72	71	adantr 486	. . . . . . . 8 ⊢ ((𝑓 = ⟨“𝑖𝑗”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → ((♯‘𝑓) = 2 ↔ (♯‘⟨“𝑖𝑗”⟩) = 2))
73	70, 72	anbi12d 644	. . . . . . 7 ⊢ ((𝑓 = ⟨“𝑖𝑗”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → ((𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (♯‘𝑓) = 2) ↔ (⟨“𝑖𝑗”⟩(𝐴(SPathsOn‘𝐺)𝐶)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘⟨“𝑖𝑗”⟩) = 2)))
74	73	spc2egv 3553	. . . . . 6 ⊢ ((⟨“𝑖𝑗”⟩ ∈ V ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ V) → ((⟨“𝑖𝑗”⟩(𝐴(SPathsOn‘𝐺)𝐶)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘⟨“𝑖𝑗”⟩) = 2) → ∃𝑓∃𝑝(𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (♯‘𝑓) = 2)))
75	25, 69, 74	mpsyl 69	. . . . 5 ⊢ (((((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) ∧ (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶})) → ∃𝑓∃𝑝(𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (♯‘𝑓) = 2))
76	75	ex 418	. . . 4 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑗 ∈ dom (iEdg‘𝐺))) → ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → ∃𝑓∃𝑝(𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (♯‘𝑓) = 2)))
77	76	rexlimdvva 3219	. . 3 ⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (∃𝑖 ∈ dom (iEdg‘𝐺)∃𝑗 ∈ dom (iEdg‘𝐺)(((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → ∃𝑓∃𝑝(𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (♯‘𝑓) = 2)))
78	20, 77	sylbid 243	. 2 ⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓∃𝑝(𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (♯‘𝑓) = 2)))
79	78	3impia 1135	1 ⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)) → ∃𝑓∃𝑝(𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (♯‘𝑓) = 2))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895   ∖ cdif 4045  ∅c0 4278  𝒫 cpw 4556  {csn 4584  {cpr 4586   class class class wbr 5102  dom cdm 5650  ran crn 5651   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420  2c2 12400  ♯chash 14477   ++ cconcat 14718  ⟨“cs1 14745  ⟨“cs2 14996  ⟨“cs3 14997  Vtxcvtx 29579  iEdgciedg 29580  Edgcedg 29630  UHGraphcuhgr 29639  SPathsOncspthson 30303

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-er 8720  df-map 8852  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-card 10023  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-nn 12339  df-2 12408  df-3 12409  df-n0 12610  df-z 12697  df-uz 12969  df-fz 13643  df-fzo 13792  df-hash 14478  df-word 14662  df-concat 14719  df-s1 14746  df-s2 15003  df-s3 15004  df-edg 29631  df-uhgr 29641  df-wlks 30185  df-wlkson 30186  df-trls 30280  df-trlson 30281  df-spths 30305  df-spthson 30307

This theorem is used by:  2pthfrgr  30890