uhgr3cyclex - Metamath Proof Explorer

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Theorem uhgr3cyclex 30813

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 2856	. . . . . 6 ⊢ ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ (Edg‘𝐺))
3		eqid 2763	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4	3	uhgredgiedgb 29734	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → ({𝐴, 𝐵} ∈ (Edg‘𝐺) ↔ ∃𝑖 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)))
5	2, 4	bitrid 286	. . . . 5 ⊢ (𝐺 ∈ UHGraph → ({𝐴, 𝐵} ∈ 𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)))
6	1	eleq2i 2856	. . . . . 6 ⊢ ({𝐵, 𝐶} ∈ 𝐸 ↔ {𝐵, 𝐶} ∈ (Edg‘𝐺))
7	3	uhgredgiedgb 29734	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → ({𝐵, 𝐶} ∈ (Edg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)))
8	6, 7	bitrid 286	. . . . 5 ⊢ (𝐺 ∈ UHGraph → ({𝐵, 𝐶} ∈ 𝐸 ↔ ∃𝑗 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)))
9	1	eleq2i 2856	. . . . . 6 ⊢ ({𝐶, 𝐴} ∈ 𝐸 ↔ {𝐶, 𝐴} ∈ (Edg‘𝐺))
10	3	uhgredgiedgb 29734	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → ({𝐶, 𝐴} ∈ (Edg‘𝐺) ↔ ∃𝑘 ∈ dom (iEdg‘𝐺){𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)))
11	9, 10	bitrid 286	. . . . 5 ⊢ (𝐺 ∈ UHGraph → ({𝐶, 𝐴} ∈ 𝐸 ↔ ∃𝑘 ∈ dom (iEdg‘𝐺){𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)))
12	5, 8, 11	3anbi123d 1465	. . . 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 486	. . 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 2763	. . . . . . . . . . . . . 14 ⊢ ⟨“𝐴𝐵𝐶𝐴”⟩ = ⟨“𝐴𝐵𝐶𝐴”⟩
15		eqid 2763	. . . . . . . . . . . . . 14 ⊢ ⟨“𝑖𝑗𝑘”⟩ = ⟨“𝑖𝑗𝑘”⟩
16		3simpa 1166	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))
17		pm3.22 465	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉))
18	17	3adant2 1149	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉))
19	16, 18	jca 521	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)))
20	19	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)))
21	20	ad2antlr 740	. . . . . . . . . . . . . 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 1166	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶))
23		necom 3012	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
24	23	biimpi 219	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ≠ 𝐵 → 𝐵 ≠ 𝐴)
25	24	anim1ci 628	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) → (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴))
26	25	3adant2 1149	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴))
27		necom 3012	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐶 ≠ 𝐴)
28	27	biimpi 219	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ≠ 𝐶 → 𝐶 ≠ 𝐴)
29	28	3ad2ant2 1152	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝐶 ≠ 𝐴)
30	22, 26, 29	3jca 1146	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴) ∧ 𝐶 ≠ 𝐴))
31	30	adantl 487	. . . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . 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 3971	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖) → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
34	33	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑖)) → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
35	34	3ad2ant3 1153	. . . . . . . . . . . . . . . 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 3971	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗) → {𝐵, 𝐶} ⊆ ((iEdg‘𝐺)‘𝑗))
37	36	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ dom (iEdg‘𝐺) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑗)) → {𝐵, 𝐶} ⊆ ((iEdg‘𝐺)‘𝑗))
38	37	3ad2ant1 1151	. . . . . . . . . . . . . . . 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 3971	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘) → {𝐶, 𝐴} ⊆ ((iEdg‘𝐺)‘𝑘))
40	39	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ dom (iEdg‘𝐺) ∧ {𝐶, 𝐴} = ((iEdg‘𝐺)‘𝑘)) → {𝐶, 𝐴} ⊆ ((iEdg‘𝐺)‘𝑘))
41	40	3ad2ant2 1152	. . . . . . . . . . . . . . . 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 1146	. . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . 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 1156	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ 𝑉)
46		simp1 1154	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐴 ∈ 𝑉)
47	45, 46	jca 521	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉))
48	47, 29	anim12i 625	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ((𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ 𝐶 ≠ 𝐴))
49	48	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) → ((𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ 𝐶 ≠ 𝐴))
50		pm3.22 465	. . . . . . . . . . . . . . . . 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 1149	. . . . . . . . . . . . . . . 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 30812	. . . . . . . . . . . . . . . 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 608	. . . . . . . . . . . . . . 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 1168	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
55		simp3 1156	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐶)
56	54, 55	anim12i 625	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐵 ≠ 𝐶))
57	56	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ UHGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))) → ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐵 ≠ 𝐶))
58		3simpc 1168	. . . . . . . . . . . . . . . 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 30812	. . . . . . . . . . . . . . . . 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 3014	. . . . . . . . . . . . . . . 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 608	. . . . . . . . . . . . . . 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 30812	. . . . . . . . . . . . . . . . . . . . . . . 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 425	. . . . . . . . . . . . . . . . . . . . . . 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 1150	. . . . . . . . . . . . . . . . . . . . . 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 33	. . . . . . . . . . . . . . . . . . . . 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 1151	. . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . 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 33	. . . . . . . . . . . . . . . . 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 1150	. . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . 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 1146	. . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . 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 30810	. . . . . . . . . . . . 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 15060	. . . . . . . . . . . . . . 15 ⊢ ⟨“𝑖𝑗𝑘”⟩ ∈ Word V
76	75	elexi 3476	. . . . . . . . . . . . . 14 ⊢ ⟨“𝑖𝑗𝑘”⟩ ∈ V
77		s4cli 15061	. . . . . . . . . . . . . . 15 ⊢ ⟨“𝐴𝐵𝐶𝐴”⟩ ∈ Word V
78	77	elexi 3476	. . . . . . . . . . . . . 14 ⊢ ⟨“𝐴𝐵𝐶𝐴”⟩ ∈ V
79		breq12 5111	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = ⟨“𝑖𝑗𝑘”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶𝐴”⟩) → (𝑓(Cycles‘𝐺)𝑝 ↔ ⟨“𝑖𝑗𝑘”⟩(Cycles‘𝐺)⟨“𝐴𝐵𝐶𝐴”⟩))
80		fveqeq2 6917	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ⟨“𝑖𝑗𝑘”⟩ → ((♯‘𝑓) = 3 ↔ (♯‘⟨“𝑖𝑗𝑘”⟩) = 3))
81	80	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = ⟨“𝑖𝑗𝑘”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶𝐴”⟩) → ((♯‘𝑓) = 3 ↔ (♯‘⟨“𝑖𝑗𝑘”⟩) = 3))
82		fveq1 6907	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶𝐴”⟩ → (𝑝‘0) = (⟨“𝐴𝐵𝐶𝐴”⟩‘0))
83	82	eqeq1d 2765	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶𝐴”⟩ → ((𝑝‘0) = 𝐴 ↔ (⟨“𝐴𝐵𝐶𝐴”⟩‘0) = 𝐴))
84	83	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = ⟨“𝑖𝑗𝑘”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶𝐴”⟩) → ((𝑝‘0) = 𝐴 ↔ (⟨“𝐴𝐵𝐶𝐴”⟩‘0) = 𝐴))
85	79, 81, 84	3anbi123d 1465	. . . . . . . . . . . . . 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 3565	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 419	. . . . . . . . . . 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 1137	. . . . . . . . . 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 3159	. . . . . . . . 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 33	. . . . . . . 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 3159	. . . . . . 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 89	. . . . . 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 3159	. . . . 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 1128	. . . 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 33	. . 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 243	. 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 1135	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∃wex 1812 ∈ wcel 2147 ≠ wne 2959 ∃wrex 3090 Vcvv 3454 ⊆ wss 3900 {cpr 4590 class class class wbr 5106 dom cdm 5655 ‘cfv 6558 0cc0 11227 3c3 12425 ♯chash 14501 Word cword 14685 ⟨“cs3 15021 ⟨“cs4 15022 Vtxcvtx 29604 iEdgciedg 29605 Edgcedg 29655 UHGraphcuhgr 29664 Cyclesccycls 30402

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304

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 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7902 df-1st 8025 df-2nd 8026 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-er 8745 df-map 8877 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-card 10047 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-n0 12634 df-z 12721 df-uz 12993 df-fz 13667 df-fzo 13816 df-hash 14502 df-word 14686 df-concat 14743 df-s1 14770 df-s2 15027 df-s3 15028 df-s4 15029 df-edg 29656 df-uhgr 29666 df-wlks 30210 df-trls 30305 df-pths 30329 df-cycls 30404

This theorem is used by: umgr3cyclex 30814

W3C validator