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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 → (∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) → ∃𝑓𝑝(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
StepHypRef Expression
1 df-rex 2902 . . 3 (∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) ↔ ∃𝑎(𝑎𝑉 ∧ ∃𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
2 df-rex 2902 . . . . . 6 (∃𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) ↔ ∃𝑏(𝑏𝑉 ∧ ∃𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
3 df-rex 2902 . . . . . . . 8 (∃𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) ↔ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
43anbi2i 726 . . . . . . 7 ((𝑏𝑉 ∧ ∃𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ↔ (𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))
54exbii 1764 . . . . . 6 (∃𝑏(𝑏𝑉 ∧ ∃𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ↔ ∃𝑏(𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))
62, 5bitri 263 . . . . 5 (∃𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) ↔ ∃𝑏(𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))
76anbi2i 726 . . . 4 ((𝑎𝑉 ∧ ∃𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ↔ (𝑎𝑉 ∧ ∃𝑏(𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))
87exbii 1764 . . 3 (∃𝑎(𝑎𝑉 ∧ ∃𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ↔ ∃𝑎(𝑎𝑉 ∧ ∃𝑏(𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))
91, 8bitri 263 . 2 (∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) ↔ ∃𝑎(𝑎𝑉 ∧ ∃𝑏(𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))
10 19.41v 1901 . . . 4 (∃𝑎((𝑎𝑉 ∧ ∃𝑏(𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))) ∧ 𝑉 USGrph 𝐸) ↔ (∃𝑎(𝑎𝑉 ∧ ∃𝑏(𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))) ∧ 𝑉 USGrph 𝐸))
11 ancom 465 . . . . . . . . 9 ((𝑎𝑉 ∧ ∃𝑏(𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))) ↔ (∃𝑏(𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ 𝑎𝑉))
12 19.41v 1901 . . . . . . . . 9 (∃𝑏((𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ 𝑎𝑉) ↔ (∃𝑏(𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ 𝑎𝑉))
1311, 12bitr4i 266 . . . . . . . 8 ((𝑎𝑉 ∧ ∃𝑏(𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))) ↔ ∃𝑏((𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ 𝑎𝑉))
1413anbi1i 727 . . . . . . 7 (((𝑎𝑉 ∧ ∃𝑏(𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))) ∧ 𝑉 USGrph 𝐸) ↔ (∃𝑏((𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ 𝑎𝑉) ∧ 𝑉 USGrph 𝐸))
15 19.41v 1901 . . . . . . 7 (∃𝑏(((𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ 𝑎𝑉) ∧ 𝑉 USGrph 𝐸) ↔ (∃𝑏((𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ 𝑎𝑉) ∧ 𝑉 USGrph 𝐸))
16 anass 679 . . . . . . . . 9 ((((𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ 𝑎𝑉) ∧ 𝑉 USGrph 𝐸) ↔ ((𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ (𝑎𝑉𝑉 USGrph 𝐸)))
17 ancom 465 . . . . . . . . . . . 12 ((𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ↔ (∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏𝑉))
18 19.41v 1901 . . . . . . . . . . . 12 (∃𝑐((𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏𝑉) ↔ (∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏𝑉))
1917, 18bitr4i 266 . . . . . . . . . . 11 ((𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ↔ ∃𝑐((𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏𝑉))
2019anbi1i 727 . . . . . . . . . 10 (((𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ (𝑎𝑉𝑉 USGrph 𝐸)) ↔ (∃𝑐((𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏𝑉) ∧ (𝑎𝑉𝑉 USGrph 𝐸)))
21 19.41v 1901 . . . . . . . . . 10 (∃𝑐(((𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏𝑉) ∧ (𝑎𝑉𝑉 USGrph 𝐸)) ↔ (∃𝑐((𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏𝑉) ∧ (𝑎𝑉𝑉 USGrph 𝐸)))
2220, 21bitr4i 266 . . . . . . . . 9 (((𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ (𝑎𝑉𝑉 USGrph 𝐸)) ↔ ∃𝑐(((𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏𝑉) ∧ (𝑎𝑉𝑉 USGrph 𝐸)))
2316, 22bitri 263 . . . . . . . 8 ((((𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ 𝑎𝑉) ∧ 𝑉 USGrph 𝐸) ↔ ∃𝑐(((𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏𝑉) ∧ (𝑎𝑉𝑉 USGrph 𝐸)))
2423exbii 1764 . . . . . . 7 (∃𝑏(((𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ∧ 𝑎𝑉) ∧ 𝑉 USGrph 𝐸) ↔ ∃𝑏𝑐(((𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏𝑉) ∧ (𝑎𝑉𝑉 USGrph 𝐸)))
2514, 15, 243bitr2i 287 . . . . . 6 (((𝑎𝑉 ∧ ∃𝑏(𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))) ∧ 𝑉 USGrph 𝐸) ↔ ∃𝑏𝑐(((𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏𝑉) ∧ (𝑎𝑉𝑉 USGrph 𝐸)))
2625exbii 1764 . . . . 5 (∃𝑎((𝑎𝑉 ∧ ∃𝑏(𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))) ∧ 𝑉 USGrph 𝐸) ↔ ∃𝑎𝑏𝑐(((𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏𝑉) ∧ (𝑎𝑉𝑉 USGrph 𝐸)))
27 simprr 792 . . . . . . . . 9 ((((𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏𝑉) ∧ (𝑎𝑉𝑉 USGrph 𝐸)) → 𝑉 USGrph 𝐸)
28 simprl 790 . . . . . . . . 9 ((((𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏𝑉) ∧ (𝑎𝑉𝑉 USGrph 𝐸)) → 𝑎𝑉)
29 simplr 788 . . . . . . . . 9 ((((𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏𝑉) ∧ (𝑎𝑉𝑉 USGrph 𝐸)) → 𝑏𝑉)
30 simplll 794 . . . . . . . . 9 ((((𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏𝑉) ∧ (𝑎𝑉𝑉 USGrph 𝐸)) → 𝑐𝑉)
31 simplr 788 . . . . . . . . . 10 (((𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏𝑉) → ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))
3231adantr 480 . . . . . . . . 9 ((((𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏𝑉) ∧ (𝑎𝑉𝑉 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⟩})
3533, 34constr3cycl 26189 . . . . . . . . 9 ((𝑉 USGrph 𝐸 ∧ (𝑎𝑉𝑏𝑉𝑐𝑉) ∧ ({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))
3627, 28, 29, 30, 32, 35syl131anc 1331 . . . . . . . 8 ((((𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏𝑉) ∧ (𝑎𝑉𝑉 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))
3736eximi 1752 . . . . . . 7 (∃𝑐(((𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏𝑉) ∧ (𝑎𝑉𝑉 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))
38372eximi 1753 . . . . . 6 (∃𝑎𝑏𝑐(((𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏𝑉) ∧ (𝑎𝑉𝑉 USGrph 𝐸)) → ∃𝑎𝑏𝑐({⟨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
4240, 41unex 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})⟩}))
4544eqeq1d 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))
4645adantr 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))
4743, 46anbi12d 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)))
4839, 42, 47spc2ev 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) → ∃𝑓𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3))
4948exlimiv 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) → ∃𝑓𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3))
5049exlimivv 1847 . . . . . 6 (∃𝑎𝑏𝑐({⟨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) → ∃𝑓𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3))
5138, 50syl 17 . . . . 5 (∃𝑎𝑏𝑐(((𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ∧ 𝑏𝑉) ∧ (𝑎𝑉𝑉 USGrph 𝐸)) → ∃𝑓𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3))
5226, 51sylbi 206 . . . 4 (∃𝑎((𝑎𝑉 ∧ ∃𝑏(𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))) ∧ 𝑉 USGrph 𝐸) → ∃𝑓𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3))
5310, 52sylbir 224 . . 3 ((∃𝑎(𝑎𝑉 ∧ ∃𝑏(𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))) ∧ 𝑉 USGrph 𝐸) → ∃𝑓𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3))
5453expcom 450 . 2 (𝑉 USGrph 𝐸 → (∃𝑎(𝑎𝑉 ∧ ∃𝑏(𝑏𝑉 ∧ ∃𝑐(𝑐𝑉 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))) → ∃𝑓𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3)))
559, 54syl5bi 231 1 (𝑉 USGrph 𝐸 → (∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) → ∃𝑓𝑝(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
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