Theorem 2pthon3v 28209

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 27322	. . . . . . . . . 10 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
3	1, 2	eqtri 2766	. . . . . . . . 9 ⊢ 𝐸 = ran (iEdg‘𝐺)
4	3	eleq2i 2830	. . . . . . . 8 ⊢ ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ ran (iEdg‘𝐺))
5		2pthon3v.v	. . . . . . . . . . 11 ⊢ 𝑉 = (Vtx‘𝐺)
6		eqid 2738	. . . . . . . . . . 11 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
7	5, 6	uhgrf 27335	. . . . . . . . . 10 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
8	7	ffnd 6585	. . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
9		fvelrnb 6812	. . . . . . . . 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 17	. . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → ({𝐴, 𝐵} ∈ ran (iEdg‘𝐺) ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵}))
11	4, 10	syl5bb 282	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → ({𝐴, 𝐵} ∈ 𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵}))
12	3	eleq2i 2830	. . . . . . . 8 ⊢ ({𝐵, 𝐶} ∈ 𝐸 ↔ {𝐵, 𝐶} ∈ ran (iEdg‘𝐺))
13		fvelrnb 6812	. . . . . . . . 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 17	. . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → ({𝐵, 𝐶} ∈ ran (iEdg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}))
15	12, 14	syl5bb 282	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → ({𝐵, 𝐶} ∈ 𝐸 ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}))
16	11, 15	anbi12d 630	. . . . . 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 480	. . . . 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 480	. . . 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 3292	. . . 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 288	. . 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 14489	. . . . . . . 8 ⊢ ⟨“𝑖𝑗”⟩ = (⟨“𝑖”⟩ ++ ⟨“𝑗”⟩)
22	21	ovexi 7289	. . . . . . 7 ⊢ ⟨“𝑖𝑗”⟩ ∈ V
23		df-s3 14490	. . . . . . . 8 ⊢ ⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩)
24	23	ovexi 7289	. . . . . . 7 ⊢ ⟨“𝐴𝐵𝐶”⟩ ∈ V
25	22, 24	pm3.2i 470	. . . . . 6 ⊢ (⟨“𝑖𝑗”⟩ ∈ V ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ V)
26		eqid 2738	. . . . . . . 8 ⊢ ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐵𝐶”⟩
27		eqid 2738	. . . . . . . 8 ⊢ ⟨“𝑖𝑗”⟩ = ⟨“𝑖𝑗”⟩
28		simp-4r 780	. . . . . . . 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 1147	. . . . . . . . 9 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶))
30	29	ad3antlr 727	. . . . . . . 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 3974	. . . . . . . . . 10 ⊢ (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
32		eqimss2 3974	. . . . . . . . . 10 ⊢ (((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶} → {𝐵, 𝐶} ⊆ ((iEdg‘𝐺)‘𝑗))
33	31, 32	anim12i 612	. . . . . . . . 9 ⊢ ((((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖) ∧ {𝐵, 𝐶} ⊆ ((iEdg‘𝐺)‘𝑗)))
34	33	adantl 481	. . . . . . . 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 6765	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (((iEdg‘𝐺)‘𝑖) = {𝐴, 𝐵} ↔ ((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵}))
36	35	anbi1d 629	. . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . 14 ⊢ ((((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘𝑗) = {𝐵, 𝐶}) → {𝐴, 𝐵} = {𝐵, 𝐶})
38		3simpa 1146	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))
39		3simpc 1148	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
40		preq12bg 4781	. . . . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐵))))
42		eqneqall 2953	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝑖 ≠ 𝑗))
43	42	com12 32	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 ≠ 𝐵 → (𝐴 = 𝐵 → 𝑖 ≠ 𝑗))
44	43	3ad2ant1 1131	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → (𝐴 = 𝐵 → 𝑖 ≠ 𝑗))
45	44	com12 32	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 = 𝐵 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝑖 ≠ 𝑗))
46	45	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝑖 ≠ 𝑗))
47		eqneqall 2953	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 = 𝐶 → (𝐴 ≠ 𝐶 → 𝑖 ≠ 𝑗))
48	47	com12 32	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 ≠ 𝐶 → (𝐴 = 𝐶 → 𝑖 ≠ 𝑗))
49	48	3ad2ant2 1132	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → (𝐴 = 𝐶 → 𝑖 ≠ 𝑗))
50	49	com12 32	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 = 𝐶 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝑖 ≠ 𝑗))
51	50	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐵) → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝑖 ≠ 𝑗))
52	46, 51	jaoi 853	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐵)) → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝑖 ≠ 𝑗))
53	41, 52	syl6bi 252	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ({𝐴, 𝐵} = {𝐵, 𝐶} → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝑖 ≠ 𝑗)))
54	53	com23 86	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 𝑖 ≠ 𝑗)))
55	54	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 𝑖 ≠ 𝑗)))
56	55	imp 406	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 𝑖 ≠ 𝑗))
57	56	com12 32	. . . . . . . . . . . . . 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 17	. . . . . . . . . . . . 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	syl6bi 252	. . . . . . . . . . . 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 86	. . . . . . . . . . 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 28	. . . . . . . . . . 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 3027	. . . . . . . . . 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 480	. . . . . . . . 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 406	. . . . . . . 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 1214	. . . . . . . . 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 480	. . . . . . . 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 28208	. . . . . . 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 14530	. . . . . . 7 ⊢ (♯‘⟨“𝑖𝑗”⟩) = 2
69	67, 68	jctir 520	. . . . . 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 5075	. . . . . . . 8 ⊢ ((𝑓 = ⟨“𝑖𝑗”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → (𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ↔ ⟨“𝑖𝑗”⟩(𝐴(SPathsOn‘𝐺)𝐶)⟨“𝐴𝐵𝐶”⟩))
71		fveqeq2 6765	. . . . . . . . 9 ⊢ (𝑓 = ⟨“𝑖𝑗”⟩ → ((♯‘𝑓) = 2 ↔ (♯‘⟨“𝑖𝑗”⟩) = 2))
72	71	adantr 480	. . . . . . . 8 ⊢ ((𝑓 = ⟨“𝑖𝑗”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → ((♯‘𝑓) = 2 ↔ (♯‘⟨“𝑖𝑗”⟩) = 2))
73	70, 72	anbi12d 630	. . . . . . 7 ⊢ ((𝑓 = ⟨“𝑖𝑗”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → ((𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (♯‘𝑓) = 2) ↔ (⟨“𝑖𝑗”⟩(𝐴(SPathsOn‘𝐺)𝐶)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘⟨“𝑖𝑗”⟩) = 2)))
74	73	spc2egv 3528	. . . . . 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 68	. . . . 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 412	. . . 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 3222	. . 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 239	. 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 1115	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))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064  Vcvv 3422   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  {csn 4558  {cpr 4560   class class class wbr 5070  dom cdm 5580  ran crn 5581   Fn wfn 6413  ‘cfv 6418  (class class class)co 7255  2c2 11958  ♯chash 13972   ++ cconcat 14201  ⟨“cs1 14228  ⟨“cs2 14482  ⟨“cs3 14483  Vtxcvtx 27269  iEdgciedg 27270  Edgcedg 27320  UHGraphcuhgr 27329  SPathsOncspthson 27984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312  df-hash 13973  df-word 14146  df-concat 14202  df-s1 14229  df-s2 14489  df-s3 14490  df-edg 27321  df-uhgr 27331  df-wlks 27869  df-wlkson 27870  df-trls 27962  df-trlson 27963  df-pths 27985  df-spths 27986  df-spthson 27988

This theorem is referenced by:  2pthfrgr  28549