uhgr3cyclex - Metamath Proof Explorer

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Theorem uhgr3cyclex 30277

Description: If there are three different vertices in a hypergraph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 12-Feb-2021.)

**Hypotheses**
Ref	Expression
uhgr3cyclex.v	⊢ 𝑉 = (Vtx‘𝐺)
uhgr3cyclex.e	⊢ 𝐸 = (Edg‘𝐺)

**Assertion**
Ref	Expression
uhgr3cyclex	⊢ ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴))

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 uhgr3cyclex Dummy variables 𝑖 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uhgr3cyclex.e	. . . . . . 7 ⊢ 𝐸 = (Edg‘𝐺)
2	1	eleq2i 2832	. . . . . 6 ⊢ ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ (Edg‘𝐺))
3		eqid 2740	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4	3	uhgredgiedgb 29220	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → ({𝐴, 𝐵} ∈ (Edg‘𝐺) ↔ ∃𝑖 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)))
5	2, 4	bitrid 284	. . . . 5 ⊢ (𝐺 ∈ UHGraph → ({𝐴, 𝐵} ∈ 𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)))
6	1	eleq2i 2832	. . . . . 6 ⊢ ({𝐵, 𝐶} ∈ 𝐸 ↔ {𝐵, 𝐶} ∈ (Edg‘𝐺))
7	3	uhgredgiedgb 29220	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → ({𝐵, 𝐶} ∈ (Edg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)))
8	6, 7	bitrid 284	. . . . 5 ⊢ (𝐺 ∈ UHGraph → ({𝐵, 𝐶} ∈ 𝐸 ↔ ∃𝑗 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)))
9	1	eleq2i 2832	. . . . . 6 ⊢ ({𝐶, 𝐴} ∈ 𝐸 ↔ {𝐶, 𝐴} ∈ (Edg‘𝐺))
10	3	uhgredgiedgb 29220	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → ({𝐶, 𝐴} ∈ (Edg‘𝐺) ↔ ∃𝑘 ∈ dom (iEdg‘𝐺){𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)))
11	9, 10	bitrid 284	. . . . 5 ⊢ (𝐺 ∈ UHGraph → ({𝐶, 𝐴} ∈ 𝐸 ↔ ∃𝑘 ∈ dom (iEdg‘𝐺){𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)))
12	5, 8, 11	3anbi123d 1444	. . . 4 ⊢ (𝐺 ∈ UHGraph → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ (∃𝑖 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖) ∧ ∃𝑗 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗) ∧ ∃𝑘 ∈ dom (iEdg‘𝐺){𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘))))
13	12	adantr 481	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ (∃𝑖 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖) ∧ ∃𝑗 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗) ∧ ∃𝑘 ∈ dom (iEdg‘𝐺){𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘))))
14		eqid 2740	. . . . . . . . . . . . . 14 ⊢ ⟨“𝐴𝐵𝐶𝐴”⟩ = ⟨“𝐴𝐵𝐶𝐴”⟩
15		eqid 2740	. . . . . . . . . . . . . 14 ⊢ ⟨“𝑖𝑗𝑘”⟩ = ⟨“𝑖𝑗𝑘”⟩
16		3simpa 1154	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))
17		pm3.22 460	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉))
18	17	3adant2 1137	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉))
19	16, 18	jca 516	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)))
20	19	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)))
21	20	ad2antlr 733	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) ∧ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)))) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)))
22		3simpa 1154	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶))
23		necom 2988	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
24	23	biimpi 217	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ≠ 𝐵 → 𝐵 ≠ 𝐴)
25	24	anim1ci 622	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) → (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴))
26	25	3adant2 1137	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴))
27		necom 2988	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐶 ≠ 𝐴)
28	27	biimpi 217	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ≠ 𝐶 → 𝐶 ≠ 𝐴)
29	28	3ad2ant2 1140	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝐶 ≠ 𝐴)
30	22, 26, 29	3jca 1134	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴) ∧ 𝐶 ≠ 𝐴))
31	30	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴) ∧ 𝐶 ≠ 𝐴))
32	31	ad2antlr 733	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) ∧ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)))) → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴) ∧ 𝐶 ≠ 𝐴))
33		eqimss 3980	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖) → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
34	33	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)) → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
35	34	3ad2ant3 1141	. . . . . . . . . . . . . . . 16 ⊢ (((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖))) → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
36		eqimss 3980	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗) → {𝐵, 𝐶} ⊆ ((iEdg‘𝐺)‘𝑗))
37	36	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) → {𝐵, 𝐶} ⊆ ((iEdg‘𝐺)‘𝑗))
38	37	3ad2ant1 1139	. . . . . . . . . . . . . . . 16 ⊢ (((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖))) → {𝐵, 𝐶} ⊆ ((iEdg‘𝐺)‘𝑗))
39		eqimss 3980	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘) → {𝐶, 𝐴} ⊆ ((iEdg‘𝐺)‘𝑘))
40	39	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) → {𝐶, 𝐴} ⊆ ((iEdg‘𝐺)‘𝑘))
41	40	3ad2ant2 1140	. . . . . . . . . . . . . . . 16 ⊢ (((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖))) → {𝐶, 𝐴} ⊆ ((iEdg‘𝐺)‘𝑘))
42	35, 38, 41	3jca 1134	. . . . . . . . . . . . . . 15 ⊢ (((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖))) → ({𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖) ∧ {𝐵, 𝐶} ⊆ ((iEdg‘𝐺)‘𝑗) ∧ {𝐶, 𝐴} ⊆ ((iEdg‘𝐺)‘𝑘)))
43	42	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) ∧ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)))) → ({𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖) ∧ {𝐵, 𝐶} ⊆ ((iEdg‘𝐺)‘𝑗) ∧ {𝐶, 𝐴} ⊆ ((iEdg‘𝐺)‘𝑘)))
44		uhgr3cyclex.v	. . . . . . . . . . . . . 14 ⊢ 𝑉 = (Vtx‘𝐺)
45		simp3 1144	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ 𝑉)
46		simp1 1142	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐴 ∈ 𝑉)
47	45, 46	jca 516	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉))
48	47, 29	anim12i 619	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ((𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ 𝐶 ≠ 𝐴))
49	48	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) → ((𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ 𝐶 ≠ 𝐴))
50		pm3.22 460	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖))) → ((𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗))))
51	50	3adant2 1137	. . . . . . . . . . . . . . . 16 ⊢ (((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖))) → ((𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗))))
52	44, 1, 3	uhgr3cyclexlem 30276	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ 𝐶 ≠ 𝐴) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)) ∧ (𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)))) → 𝑖 ≠ 𝑗)
53	49, 51, 52	syl2an 602	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) ∧ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)))) → 𝑖 ≠ 𝑗)
54		3simpc 1156	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
55		simp3 1144	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐶)
56	54, 55	anim12i 619	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐵 ≠ 𝐶))
57	56	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) → ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐵 ≠ 𝐶))
58		3simpc 1156	. . . . . . . . . . . . . . . 16 ⊢ (((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖))) → ((𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖))))
59	44, 1, 3	uhgr3cyclexlem 30276	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐵 ≠ 𝐶) ∧ ((𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)))) → 𝑘 ≠ 𝑖)
60	59	necomd 2990	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐵 ≠ 𝐶) ∧ ((𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)))) → 𝑖 ≠ 𝑘)
61	57, 58, 60	syl2an 602	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) ∧ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)))) → 𝑖 ≠ 𝑘)
62	44, 1, 3	uhgr3cyclexlem 30276	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)))) → 𝑗 ≠ 𝑘)
63	62	exp31 420	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 ≠ 𝐵 → (((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘))) → 𝑗 ≠ 𝑘)))
64	63	3adant3 1138	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 ≠ 𝐵 → (((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘))) → 𝑗 ≠ 𝑘)))
65	64	com12 32	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ≠ 𝐵 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘))) → 𝑗 ≠ 𝑘)))
66	65	3ad2ant1 1139	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘))) → 𝑗 ≠ 𝑘)))
67	66	impcom 408	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘))) → 𝑗 ≠ 𝑘))
68	67	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) → (((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘))) → 𝑗 ≠ 𝑘))
69	68	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘))) → ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) → 𝑗 ≠ 𝑘))
70	69	3adant3 1138	. . . . . . . . . . . . . . . 16 ⊢ (((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖))) → ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) → 𝑗 ≠ 𝑘))
71	70	impcom 408	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) ∧ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)))) → 𝑗 ≠ 𝑘)
72	53, 61, 71	3jca 1134	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) ∧ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)))) → (𝑖 ≠ 𝑗 ∧ 𝑖 ≠ 𝑘 ∧ 𝑗 ≠ 𝑘))
73		eqidd 2741	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) ∧ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)))) → 𝐴 = 𝐴)
74	14, 15, 21, 32, 43, 44, 3, 72, 73	3cyclpd 30274	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) ∧ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)))) → (⟨“𝑖𝑗𝑘”⟩(Cycles‘𝐺)⟨“𝐴𝐵𝐶𝐴”⟩ ∧ (♯‘⟨“𝑖𝑗𝑘”⟩) = 3 ∧ (⟨“𝐴𝐵𝐶𝐴”⟩‘0) = 𝐴))
75		s3cli 14841	. . . . . . . . . . . . . . 15 ⊢ ⟨“𝑖𝑗𝑘”⟩ ∈ Word V
76	75	elexi 3455	. . . . . . . . . . . . . 14 ⊢ ⟨“𝑖𝑗𝑘”⟩ ∈ V
77		s4cli 14842	. . . . . . . . . . . . . . 15 ⊢ ⟨“𝐴𝐵𝐶𝐴”⟩ ∈ Word V
78	77	elexi 3455	. . . . . . . . . . . . . 14 ⊢ ⟨“𝐴𝐵𝐶𝐴”⟩ ∈ V
79		breq12 5084	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = ⟨“𝑖𝑗𝑘”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶𝐴”⟩) → (𝑓(Cycles‘𝐺)𝑝 ↔ ⟨“𝑖𝑗𝑘”⟩(Cycles‘𝐺)⟨“𝐴𝐵𝐶𝐴”⟩))
80		fveqeq2 6843	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ⟨“𝑖𝑗𝑘”⟩ → ((♯‘𝑓) = 3 ↔ (♯‘⟨“𝑖𝑗𝑘”⟩) = 3))
81	80	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = ⟨“𝑖𝑗𝑘”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶𝐴”⟩) → ((♯‘𝑓) = 3 ↔ (♯‘⟨“𝑖𝑗𝑘”⟩) = 3))
82		fveq1 6833	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶𝐴”⟩ → (𝑝‘0) = (⟨“𝐴𝐵𝐶𝐴”⟩‘0))
83	82	eqeq1d 2742	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶𝐴”⟩ → ((𝑝‘0) = 𝐴 ↔ (⟨“𝐴𝐵𝐶𝐴”⟩‘0) = 𝐴))
84	83	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = ⟨“𝑖𝑗𝑘”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶𝐴”⟩) → ((𝑝‘0) = 𝐴 ↔ (⟨“𝐴𝐵𝐶𝐴”⟩‘0) = 𝐴))
85	79, 81, 84	3anbi123d 1444	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = ⟨“𝑖𝑗𝑘”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶𝐴”⟩) → ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴) ↔ (⟨“𝑖𝑗𝑘”⟩(Cycles‘𝐺)⟨“𝐴𝐵𝐶𝐴”⟩ ∧ (♯‘⟨“𝑖𝑗𝑘”⟩) = 3 ∧ (⟨“𝐴𝐵𝐶𝐴”⟩‘0) = 𝐴)))
86	76, 78, 85	spc2ev 3552	. . . . . . . . . . . . 13 ⊢ ((⟨“𝑖𝑗𝑘”⟩(Cycles‘𝐺)⟨“𝐴𝐵𝐶𝐴”⟩ ∧ (♯‘⟨“𝑖𝑗𝑘”⟩) = 3 ∧ (⟨“𝐴𝐵𝐶𝐴”⟩‘0) = 𝐴) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴))
87	74, 86	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) ∧ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)))) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴))
88	87	expcom 414	. . . . . . . . . . 11 ⊢ (((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) ∧ (𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖))) → ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴)))
89	88	3exp 1125	. . . . . . . . . 10 ⊢ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) → ((𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) → ((𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)) → ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴)))))
90	89	rexlimiva 3133	. . . . . . . . 9 ⊢ (∃𝑗 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗) → ((𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) → ((𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)) → ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴)))))
91	90	com12 32	. . . . . . . 8 ⊢ ((𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) → (∃𝑗 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗) → ((𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)) → ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴)))))
92	91	rexlimiva 3133	. . . . . . 7 ⊢ (∃𝑘 ∈ dom (iEdg‘𝐺){𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘) → (∃𝑗 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗) → ((𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)) → ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴)))))
93	92	com13 88	. . . . . 6 ⊢ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)) → (∃𝑗 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗) → (∃𝑘 ∈ dom (iEdg‘𝐺){𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘) → ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴)))))
94	93	rexlimiva 3133	. . . . 5 ⊢ (∃𝑖 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖) → (∃𝑗 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗) → (∃𝑘 ∈ dom (iEdg‘𝐺){𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘) → ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴)))))
95	94	3imp 1116	. . . 4 ⊢ ((∃𝑖 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖) ∧ ∃𝑗 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗) ∧ ∃𝑘 ∈ dom (iEdg‘𝐺){𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) → ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴)))
96	95	com12 32	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) → ((∃𝑖 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖) ∧ ∃𝑗 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗) ∧ ∃𝑘 ∈ dom (iEdg‘𝐺){𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴)))
97	13, 96	sylbid 241	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴)))
98	97	3impia 1123	1 ⊢ ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∃wex 1786 ∈ wcel 2119 ≠ wne 2935 ∃wrex 3064 Vcvv 3432 ⊆ wss 3890 {cpr 4564 class class class wbr 5079 dom cdm 5625 ‘cfv 6492 0cc0 11036 3c3 12235 ♯chash 14290 Word cword 14473 ⟨“cs3 14802 ⟨“cs4 14803 Vtxcvtx 29090 iEdgciedg 29091 Edgcedg 29141 UHGraphcuhgr 29150 Cyclesccycls 29878

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-ifp 1069 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-n0 12436 df-z 12523 df-uz 12787 df-fz 13460 df-fzo 13607 df-hash 14291 df-word 14474 df-concat 14531 df-s1 14557 df-s2 14808 df-s3 14809 df-s4 14810 df-edg 29142 df-uhgr 29152 df-wlks 29693 df-trls 29784 df-pths 29807 df-cycls 29880

This theorem is referenced by: umgr3cyclex 30278

W3C validator