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Theorem 3v3e3cycl2 26192

Description: If there are three (different) vertices in a graph which are mutually connected by edges, there is a 3-cycle in the graph. (Contributed by Alexander van der Vekens, 14-Nov-2017.)

**Assertion**
Ref	Expression
3v3e3cycl2	⊢ (𝑉 USGrph 𝐸 → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) → ∃𝑓∃𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3)))

Distinct variable groups: 𝐸,𝑎,𝑏,𝑐,𝑓,𝑝 𝑉,𝑎,𝑏,𝑐,𝑓,𝑝

**Proof of Theorem 3v3e3cycl2**
Step	Hyp	Ref	Expression
1		df-rex 2902	. . 3 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) ↔ ∃𝑎(𝑎 ∈ 𝑉 ∧ ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
2		df-rex 2902	. . . . . 6 ⊢ (∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) ↔ ∃𝑏(𝑏 ∈ 𝑉 ∧ ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
3		df-rex 2902	. . . . . . . 8 ⊢ (∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) ↔ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
4	3	anbi2i 726	. . . . . . 7 ⊢ ((𝑏 ∈ 𝑉 ∧ ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ↔ (𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))
5	4	exbii 1764	. . . . . 6 ⊢ (∃𝑏(𝑏 ∈ 𝑉 ∧ ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ↔ ∃𝑏(𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))
6	2, 5	bitri 263	. . . . 5 ⊢ (∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) ↔ ∃𝑏(𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))
7	6	anbi2i 726	. . . 4 ⊢ ((𝑎 ∈ 𝑉 ∧ ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ↔ (𝑎 ∈ 𝑉 ∧ ∃𝑏(𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))
8	7	exbii 1764	. . 3 ⊢ (∃𝑎(𝑎 ∈ 𝑉 ∧ ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ↔ ∃𝑎(𝑎 ∈ 𝑉 ∧ ∃𝑏(𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))
9	1, 8	bitri 263	. 2 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) ↔ ∃𝑎(𝑎 ∈ 𝑉 ∧ ∃𝑏(𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))
10		19.41v 1901	. . . 4 ⊢ (∃𝑎((𝑎 ∈ 𝑉 ∧ ∃𝑏(𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))) ∧ 𝑉 USGrph 𝐸) ↔ (∃𝑎(𝑎 ∈ 𝑉 ∧ ∃𝑏(𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))) ∧ 𝑉 USGrph 𝐸))
11		ancom 465	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝑉 ∧ ∃𝑏(𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))) ↔ (∃𝑏(𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ 𝑎 ∈ 𝑉))
12		19.41v 1901	. . . . . . . . 9 ⊢ (∃𝑏((𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ 𝑎 ∈ 𝑉) ↔ (∃𝑏(𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ 𝑎 ∈ 𝑉))
13	11, 12	bitr4i 266	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝑉 ∧ ∃𝑏(𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))) ↔ ∃𝑏((𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ 𝑎 ∈ 𝑉))
14	13	anbi1i 727	. . . . . . 7 ⊢ (((𝑎 ∈ 𝑉 ∧ ∃𝑏(𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))) ∧ 𝑉 USGrph 𝐸) ↔ (∃𝑏((𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑉 USGrph 𝐸))
15		19.41v 1901	. . . . . . 7 ⊢ (∃𝑏(((𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑉 USGrph 𝐸) ↔ (∃𝑏((𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑉 USGrph 𝐸))
16		anass 679	. . . . . . . . 9 ⊢ ((((𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑉 USGrph 𝐸) ↔ ((𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑉 USGrph 𝐸)))
17		ancom 465	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ↔ (∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏 ∈ 𝑉))
18		19.41v 1901	. . . . . . . . . . . 12 ⊢ (∃𝑐((𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏 ∈ 𝑉) ↔ (∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏 ∈ 𝑉))
19	17, 18	bitr4i 266	. . . . . . . . . . 11 ⊢ ((𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ↔ ∃𝑐((𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏 ∈ 𝑉))
20	19	anbi1i 727	. . . . . . . . . 10 ⊢ (((𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑉 USGrph 𝐸)) ↔ (∃𝑐((𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑉 USGrph 𝐸)))
21		19.41v 1901	. . . . . . . . . 10 ⊢ (∃𝑐(((𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑉 USGrph 𝐸)) ↔ (∃𝑐((𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑉 USGrph 𝐸)))
22	20, 21	bitr4i 266	. . . . . . . . 9 ⊢ (((𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑉 USGrph 𝐸)) ↔ ∃𝑐(((𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑉 USGrph 𝐸)))
23	16, 22	bitri 263	. . . . . . . 8 ⊢ ((((𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑉 USGrph 𝐸) ↔ ∃𝑐(((𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑉 USGrph 𝐸)))
24	23	exbii 1764	. . . . . . 7 ⊢ (∃𝑏(((𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑉 USGrph 𝐸) ↔ ∃𝑏∃𝑐(((𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑉 USGrph 𝐸)))
25	14, 15, 24	3bitr2i 287	. . . . . 6 ⊢ (((𝑎 ∈ 𝑉 ∧ ∃𝑏(𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))) ∧ 𝑉 USGrph 𝐸) ↔ ∃𝑏∃𝑐(((𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑉 USGrph 𝐸)))
26	25	exbii 1764	. . . . 5 ⊢ (∃𝑎((𝑎 ∈ 𝑉 ∧ ∃𝑏(𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))) ∧ 𝑉 USGrph 𝐸) ↔ ∃𝑎∃𝑏∃𝑐(((𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑉 USGrph 𝐸)))
27		simprr 792	. . . . . . . . 9 ⊢ ((((𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑉 USGrph 𝐸)) → 𝑉 USGrph 𝐸)
28		simprl 790	. . . . . . . . 9 ⊢ ((((𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑉 USGrph 𝐸)) → 𝑎 ∈ 𝑉)
29		simplr 788	. . . . . . . . 9 ⊢ ((((𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑉 USGrph 𝐸)) → 𝑏 ∈ 𝑉)
30		simplll 794	. . . . . . . . 9 ⊢ ((((𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑉 USGrph 𝐸)) → 𝑐 ∈ 𝑉)
31		simplr 788	. . . . . . . . . 10 ⊢ (((𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏 ∈ 𝑉) → ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))
32	31	adantr 480	. . . . . . . . 9 ⊢ ((((𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑉 USGrph 𝐸)) → ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))
33		eqid 2610	. . . . . . . . . 10 ⊢ {⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩} = {⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩}
34		eqid 2610	. . . . . . . . . 10 ⊢ ({⟨0, 𝑎⟩, ⟨1, 𝑏⟩} ∪ {⟨2, 𝑐⟩, ⟨3, 𝑎⟩}) = ({⟨0, 𝑎⟩, ⟨1, 𝑏⟩} ∪ {⟨2, 𝑐⟩, ⟨3, 𝑎⟩})
35	33, 34	constr3cycl 26189	. . . . . . . . 9 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → ({⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩} (𝑉 Cycles 𝐸)({⟨0, 𝑎⟩, ⟨1, 𝑏⟩} ∪ {⟨2, 𝑐⟩, ⟨3, 𝑎⟩}) ∧ (#‘{⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩}) = 3))
36	27, 28, 29, 30, 32, 35	syl131anc 1331	. . . . . . . 8 ⊢ ((((𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑉 USGrph 𝐸)) → ({⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩} (𝑉 Cycles 𝐸)({⟨0, 𝑎⟩, ⟨1, 𝑏⟩} ∪ {⟨2, 𝑐⟩, ⟨3, 𝑎⟩}) ∧ (#‘{⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩}) = 3))
37	36	eximi 1752	. . . . . . 7 ⊢ (∃𝑐(((𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑉 USGrph 𝐸)) → ∃𝑐({⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩} (𝑉 Cycles 𝐸)({⟨0, 𝑎⟩, ⟨1, 𝑏⟩} ∪ {⟨2, 𝑐⟩, ⟨3, 𝑎⟩}) ∧ (#‘{⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩}) = 3))
38	37	2eximi 1753	. . . . . 6 ⊢ (∃𝑎∃𝑏∃𝑐(((𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑉 USGrph 𝐸)) → ∃𝑎∃𝑏∃𝑐({⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩} (𝑉 Cycles 𝐸)({⟨0, 𝑎⟩, ⟨1, 𝑏⟩} ∪ {⟨2, 𝑐⟩, ⟨3, 𝑎⟩}) ∧ (#‘{⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩}) = 3))
39		tpex 6855	. . . . . . . . 9 ⊢ {⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩} ∈ V
40		prex 4836	. . . . . . . . . 10 ⊢ {⟨0, 𝑎⟩, ⟨1, 𝑏⟩} ∈ V
41		prex 4836	. . . . . . . . . 10 ⊢ {⟨2, 𝑐⟩, ⟨3, 𝑎⟩} ∈ V
42	40, 41	unex 6854	. . . . . . . . 9 ⊢ ({⟨0, 𝑎⟩, ⟨1, 𝑏⟩} ∪ {⟨2, 𝑐⟩, ⟨3, 𝑎⟩}) ∈ V
43		breq12 4588	. . . . . . . . . 10 ⊢ ((𝑓 = {⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩} ∧ 𝑝 = ({⟨0, 𝑎⟩, ⟨1, 𝑏⟩} ∪ {⟨2, 𝑐⟩, ⟨3, 𝑎⟩})) → (𝑓(𝑉 Cycles 𝐸)𝑝 ↔ {⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩} (𝑉 Cycles 𝐸)({⟨0, 𝑎⟩, ⟨1, 𝑏⟩} ∪ {⟨2, 𝑐⟩, ⟨3, 𝑎⟩})))
44		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑓 = {⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩} → (#‘𝑓) = (#‘{⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩}))
45	44	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (𝑓 = {⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩} → ((#‘𝑓) = 3 ↔ (#‘{⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩}) = 3))
46	45	adantr 480	. . . . . . . . . 10 ⊢ ((𝑓 = {⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩} ∧ 𝑝 = ({⟨0, 𝑎⟩, ⟨1, 𝑏⟩} ∪ {⟨2, 𝑐⟩, ⟨3, 𝑎⟩})) → ((#‘𝑓) = 3 ↔ (#‘{⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩}) = 3))
47	43, 46	anbi12d 743	. . . . . . . . 9 ⊢ ((𝑓 = {⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩} ∧ 𝑝 = ({⟨0, 𝑎⟩, ⟨1, 𝑏⟩} ∪ {⟨2, 𝑐⟩, ⟨3, 𝑎⟩})) → ((𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3) ↔ ({⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩} (𝑉 Cycles 𝐸)({⟨0, 𝑎⟩, ⟨1, 𝑏⟩} ∪ {⟨2, 𝑐⟩, ⟨3, 𝑎⟩}) ∧ (#‘{⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩}) = 3)))
48	39, 42, 47	spc2ev 3274	. . . . . . . 8 ⊢ (({⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩} (𝑉 Cycles 𝐸)({⟨0, 𝑎⟩, ⟨1, 𝑏⟩} ∪ {⟨2, 𝑐⟩, ⟨3, 𝑎⟩}) ∧ (#‘{⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩}) = 3) → ∃𝑓∃𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3))
49	48	exlimiv 1845	. . . . . . 7 ⊢ (∃𝑐({⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩} (𝑉 Cycles 𝐸)({⟨0, 𝑎⟩, ⟨1, 𝑏⟩} ∪ {⟨2, 𝑐⟩, ⟨3, 𝑎⟩}) ∧ (#‘{⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩}) = 3) → ∃𝑓∃𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3))
50	49	exlimivv 1847	. . . . . 6 ⊢ (∃𝑎∃𝑏∃𝑐({⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩} (𝑉 Cycles 𝐸)({⟨0, 𝑎⟩, ⟨1, 𝑏⟩} ∪ {⟨2, 𝑐⟩, ⟨3, 𝑎⟩}) ∧ (#‘{⟨0, (^◡𝐸‘{𝑎, 𝑏})⟩, ⟨1, (^◡𝐸‘{𝑏, 𝑐})⟩, ⟨2, (^◡𝐸‘{𝑐, 𝑎})⟩}) = 3) → ∃𝑓∃𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3))
51	38, 50	syl 17	. . . . 5 ⊢ (∃𝑎∃𝑏∃𝑐(((𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑉 USGrph 𝐸)) → ∃𝑓∃𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3))
52	26, 51	sylbi 206	. . . 4 ⊢ (∃𝑎((𝑎 ∈ 𝑉 ∧ ∃𝑏(𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))) ∧ 𝑉 USGrph 𝐸) → ∃𝑓∃𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3))
53	10, 52	sylbir 224	. . 3 ⊢ ((∃𝑎(𝑎 ∈ 𝑉 ∧ ∃𝑏(𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))) ∧ 𝑉 USGrph 𝐸) → ∃𝑓∃𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3))
54	53	expcom 450	. 2 ⊢ (𝑉 USGrph 𝐸 → (∃𝑎(𝑎 ∈ 𝑉 ∧ ∃𝑏(𝑏 ∈ 𝑉 ∧ ∃𝑐(𝑐 ∈ 𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))) → ∃𝑓∃𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3)))
55	9, 54	syl5bi 231	1 ⊢ (𝑉 USGrph 𝐸 → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) → ∃𝑓∃𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∃wrex 2897 ∪ cun 3538 {cpr 4127 {ctp 4129 ⟨cop 4131 class class class wbr 4583 ^◡ccnv 5037 ran crn 5039 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 2c2 10947 3c3 10948 #chash 12979 USGrph cusg 25859 Cycles ccycl 26035

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-usgra 25862 df-wlk 26036 df-trail 26037 df-pth 26038 df-cycl 26041

This theorem is referenced by: 3v3e3cycl 26193

W3C validator