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Theorem usgrgrtrirex 47728
Description: Conditions for a simple graph to contain a triangle. (Contributed by AV, 7-Aug-2025.)
Hypotheses
Ref Expression
usgrgrtrirex.v 𝑉 = (Vtx‘𝐺)
usgrgrtrirex.e 𝐸 = (Edg‘𝐺)
usgrgrtrirex.n 𝑁 = (𝐺 NeighbVtx 𝑎)
Assertion
Ref Expression
usgrgrtrirex (𝐺 ∈ USGraph → (∃𝑡 𝑡 ∈ (GrTriangles‘𝐺) ↔ ∃𝑎𝑉𝑏𝑁𝑐𝑁 (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)))
Distinct variable groups:   𝐸,𝑎,𝑏,𝑐,𝑡   𝐺,𝑎,𝑏,𝑐,𝑡   𝑁,𝑏,𝑐,𝑡   𝑉,𝑎,𝑏,𝑐,𝑡
Allowed substitution hint:   𝑁(𝑎)

Proof of Theorem usgrgrtrirex
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgrgrtrirex.v . . . 4 𝑉 = (Vtx‘𝐺)
2 usgrgrtrirex.e . . . 4 𝐸 = (Edg‘𝐺)
31, 2isgrtri 47724 . . 3 (𝑡 ∈ (GrTriangles‘𝐺) ↔ ∃𝑎𝑉𝑦𝑉𝑧𝑉 (𝑡 = {𝑎, 𝑦, 𝑧} ∧ (♯‘𝑡) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))
43exbii 1846 . 2 (∃𝑡 𝑡 ∈ (GrTriangles‘𝐺) ↔ ∃𝑡𝑎𝑉𝑦𝑉𝑧𝑉 (𝑡 = {𝑎, 𝑦, 𝑧} ∧ (♯‘𝑡) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))
5 rexcom4 3289 . . 3 (∃𝑎𝑉𝑡𝑦𝑉𝑧𝑉 (𝑡 = {𝑎, 𝑦, 𝑧} ∧ (♯‘𝑡) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) ↔ ∃𝑡𝑎𝑉𝑦𝑉𝑧𝑉 (𝑡 = {𝑎, 𝑦, 𝑧} ∧ (♯‘𝑡) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))
6 fveqeq2 6928 . . . . . . . . . . 11 (𝑡 = {𝑎, 𝑦, 𝑧} → ((♯‘𝑡) = 3 ↔ (♯‘{𝑎, 𝑦, 𝑧}) = 3))
76adantl 481 . . . . . . . . . 10 ((((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ (𝑦𝑉𝑧𝑉)) ∧ 𝑡 = {𝑎, 𝑦, 𝑧}) → ((♯‘𝑡) = 3 ↔ (♯‘{𝑎, 𝑦, 𝑧}) = 3))
8 neeq1 3005 . . . . . . . . . . . . . 14 (𝑏 = 𝑦 → (𝑏𝑐𝑦𝑐))
9 preq1 4758 . . . . . . . . . . . . . . 15 (𝑏 = 𝑦 → {𝑏, 𝑐} = {𝑦, 𝑐})
109eleq1d 2823 . . . . . . . . . . . . . 14 (𝑏 = 𝑦 → ({𝑏, 𝑐} ∈ 𝐸 ↔ {𝑦, 𝑐} ∈ 𝐸))
118, 10anbi12d 631 . . . . . . . . . . . . 13 (𝑏 = 𝑦 → ((𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸) ↔ (𝑦𝑐 ∧ {𝑦, 𝑐} ∈ 𝐸)))
12 neeq2 3006 . . . . . . . . . . . . . 14 (𝑐 = 𝑧 → (𝑦𝑐𝑦𝑧))
13 preq2 4759 . . . . . . . . . . . . . . 15 (𝑐 = 𝑧 → {𝑦, 𝑐} = {𝑦, 𝑧})
1413eleq1d 2823 . . . . . . . . . . . . . 14 (𝑐 = 𝑧 → ({𝑦, 𝑐} ∈ 𝐸 ↔ {𝑦, 𝑧} ∈ 𝐸))
1512, 14anbi12d 631 . . . . . . . . . . . . 13 (𝑐 = 𝑧 → ((𝑦𝑐 ∧ {𝑦, 𝑐} ∈ 𝐸) ↔ (𝑦𝑧 ∧ {𝑦, 𝑧} ∈ 𝐸)))
16 prcom 4757 . . . . . . . . . . . . . . . . . . . . . 22 {𝑎, 𝑦} = {𝑦, 𝑎}
1716eleq1i 2829 . . . . . . . . . . . . . . . . . . . . 21 ({𝑎, 𝑦} ∈ 𝐸 ↔ {𝑦, 𝑎} ∈ 𝐸)
182nbusgreledg 29379 . . . . . . . . . . . . . . . . . . . . . 22 (𝐺 ∈ USGraph → (𝑦 ∈ (𝐺 NeighbVtx 𝑎) ↔ {𝑦, 𝑎} ∈ 𝐸))
1918biimprcd 250 . . . . . . . . . . . . . . . . . . . . 21 ({𝑦, 𝑎} ∈ 𝐸 → (𝐺 ∈ USGraph → 𝑦 ∈ (𝐺 NeighbVtx 𝑎)))
2017, 19sylbi 217 . . . . . . . . . . . . . . . . . . . 20 ({𝑎, 𝑦} ∈ 𝐸 → (𝐺 ∈ USGraph → 𝑦 ∈ (𝐺 NeighbVtx 𝑎)))
21203ad2ant1 1133 . . . . . . . . . . . . . . . . . . 19 (({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸) → (𝐺 ∈ USGraph → 𝑦 ∈ (𝐺 NeighbVtx 𝑎)))
2221com12 32 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ USGraph → (({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸) → 𝑦 ∈ (𝐺 NeighbVtx 𝑎)))
2322adantr 480 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ USGraph ∧ 𝑎𝑉) → (({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸) → 𝑦 ∈ (𝐺 NeighbVtx 𝑎)))
2423adantr 480 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ (𝑦𝑉𝑧𝑉)) → (({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸) → 𝑦 ∈ (𝐺 NeighbVtx 𝑎)))
2524a1d 25 . . . . . . . . . . . . . . 15 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ (𝑦𝑉𝑧𝑉)) → ((♯‘{𝑎, 𝑦, 𝑧}) = 3 → (({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸) → 𝑦 ∈ (𝐺 NeighbVtx 𝑎))))
26253imp 1111 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ (𝑦𝑉𝑧𝑉)) ∧ (♯‘{𝑎, 𝑦, 𝑧}) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → 𝑦 ∈ (𝐺 NeighbVtx 𝑎))
27 usgrgrtrirex.n . . . . . . . . . . . . . 14 𝑁 = (𝐺 NeighbVtx 𝑎)
2826, 27eleqtrrdi 2849 . . . . . . . . . . . . 13 ((((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ (𝑦𝑉𝑧𝑉)) ∧ (♯‘{𝑎, 𝑦, 𝑧}) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → 𝑦𝑁)
29 prcom 4757 . . . . . . . . . . . . . . . . . . . . . 22 {𝑎, 𝑧} = {𝑧, 𝑎}
3029eleq1i 2829 . . . . . . . . . . . . . . . . . . . . 21 ({𝑎, 𝑧} ∈ 𝐸 ↔ {𝑧, 𝑎} ∈ 𝐸)
312nbusgreledg 29379 . . . . . . . . . . . . . . . . . . . . . 22 (𝐺 ∈ USGraph → (𝑧 ∈ (𝐺 NeighbVtx 𝑎) ↔ {𝑧, 𝑎} ∈ 𝐸))
3231biimprcd 250 . . . . . . . . . . . . . . . . . . . . 21 ({𝑧, 𝑎} ∈ 𝐸 → (𝐺 ∈ USGraph → 𝑧 ∈ (𝐺 NeighbVtx 𝑎)))
3330, 32sylbi 217 . . . . . . . . . . . . . . . . . . . 20 ({𝑎, 𝑧} ∈ 𝐸 → (𝐺 ∈ USGraph → 𝑧 ∈ (𝐺 NeighbVtx 𝑎)))
34333ad2ant2 1134 . . . . . . . . . . . . . . . . . . 19 (({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸) → (𝐺 ∈ USGraph → 𝑧 ∈ (𝐺 NeighbVtx 𝑎)))
3534com12 32 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ USGraph → (({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸) → 𝑧 ∈ (𝐺 NeighbVtx 𝑎)))
3635adantr 480 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ USGraph ∧ 𝑎𝑉) → (({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸) → 𝑧 ∈ (𝐺 NeighbVtx 𝑎)))
3736adantr 480 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ (𝑦𝑉𝑧𝑉)) → (({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸) → 𝑧 ∈ (𝐺 NeighbVtx 𝑎)))
3837a1d 25 . . . . . . . . . . . . . . 15 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ (𝑦𝑉𝑧𝑉)) → ((♯‘{𝑎, 𝑦, 𝑧}) = 3 → (({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸) → 𝑧 ∈ (𝐺 NeighbVtx 𝑎))))
39383imp 1111 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ (𝑦𝑉𝑧𝑉)) ∧ (♯‘{𝑎, 𝑦, 𝑧}) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → 𝑧 ∈ (𝐺 NeighbVtx 𝑎))
4039, 27eleqtrrdi 2849 . . . . . . . . . . . . 13 ((((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ (𝑦𝑉𝑧𝑉)) ∧ (♯‘{𝑎, 𝑦, 𝑧}) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → 𝑧𝑁)
41 hashtpg 14530 . . . . . . . . . . . . . . . . . 18 ((𝑎 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑎𝑦𝑦𝑧𝑧𝑎) ↔ (♯‘{𝑎, 𝑦, 𝑧}) = 3))
4241bicomd 223 . . . . . . . . . . . . . . . . 17 ((𝑎 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((♯‘{𝑎, 𝑦, 𝑧}) = 3 ↔ (𝑎𝑦𝑦𝑧𝑧𝑎)))
4342el3v 3491 . . . . . . . . . . . . . . . 16 ((♯‘{𝑎, 𝑦, 𝑧}) = 3 ↔ (𝑎𝑦𝑦𝑧𝑧𝑎))
4443simp2bi 1146 . . . . . . . . . . . . . . 15 ((♯‘{𝑎, 𝑦, 𝑧}) = 3 → 𝑦𝑧)
45443ad2ant2 1134 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ (𝑦𝑉𝑧𝑉)) ∧ (♯‘{𝑎, 𝑦, 𝑧}) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → 𝑦𝑧)
46 simp33 1211 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ (𝑦𝑉𝑧𝑉)) ∧ (♯‘{𝑎, 𝑦, 𝑧}) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → {𝑦, 𝑧} ∈ 𝐸)
4745, 46jca 511 . . . . . . . . . . . . 13 ((((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ (𝑦𝑉𝑧𝑉)) ∧ (♯‘{𝑎, 𝑦, 𝑧}) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → (𝑦𝑧 ∧ {𝑦, 𝑧} ∈ 𝐸))
4811, 15, 28, 40, 472rspcedvdw 3645 . . . . . . . . . . . 12 ((((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ (𝑦𝑉𝑧𝑉)) ∧ (♯‘{𝑎, 𝑦, 𝑧}) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → ∃𝑏𝑁𝑐𝑁 (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸))
49483exp 1119 . . . . . . . . . . 11 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ (𝑦𝑉𝑧𝑉)) → ((♯‘{𝑎, 𝑦, 𝑧}) = 3 → (({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸) → ∃𝑏𝑁𝑐𝑁 (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸))))
5049adantr 480 . . . . . . . . . 10 ((((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ (𝑦𝑉𝑧𝑉)) ∧ 𝑡 = {𝑎, 𝑦, 𝑧}) → ((♯‘{𝑎, 𝑦, 𝑧}) = 3 → (({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸) → ∃𝑏𝑁𝑐𝑁 (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸))))
517, 50sylbid 240 . . . . . . . . 9 ((((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ (𝑦𝑉𝑧𝑉)) ∧ 𝑡 = {𝑎, 𝑦, 𝑧}) → ((♯‘𝑡) = 3 → (({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸) → ∃𝑏𝑁𝑐𝑁 (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸))))
5251ex 412 . . . . . . . 8 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ (𝑦𝑉𝑧𝑉)) → (𝑡 = {𝑎, 𝑦, 𝑧} → ((♯‘𝑡) = 3 → (({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸) → ∃𝑏𝑁𝑐𝑁 (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)))))
53523impd 1348 . . . . . . 7 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ (𝑦𝑉𝑧𝑉)) → ((𝑡 = {𝑎, 𝑦, 𝑧} ∧ (♯‘𝑡) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → ∃𝑏𝑁𝑐𝑁 (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)))
5453rexlimdvva 3215 . . . . . 6 ((𝐺 ∈ USGraph ∧ 𝑎𝑉) → (∃𝑦𝑉𝑧𝑉 (𝑡 = {𝑎, 𝑦, 𝑧} ∧ (♯‘𝑡) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → ∃𝑏𝑁𝑐𝑁 (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)))
5554exlimdv 1932 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑎𝑉) → (∃𝑡𝑦𝑉𝑧𝑉 (𝑡 = {𝑎, 𝑦, 𝑧} ∧ (♯‘𝑡) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → ∃𝑏𝑁𝑐𝑁 (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)))
5627eleq2i 2830 . . . . . . . . . 10 (𝑏𝑁𝑏 ∈ (𝐺 NeighbVtx 𝑎))
572nbusgreledg 29379 . . . . . . . . . 10 (𝐺 ∈ USGraph → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ {𝑏, 𝑎} ∈ 𝐸))
5856, 57bitrid 283 . . . . . . . . 9 (𝐺 ∈ USGraph → (𝑏𝑁 ↔ {𝑏, 𝑎} ∈ 𝐸))
5927eleq2i 2830 . . . . . . . . . 10 (𝑐𝑁𝑐 ∈ (𝐺 NeighbVtx 𝑎))
602nbusgreledg 29379 . . . . . . . . . 10 (𝐺 ∈ USGraph → (𝑐 ∈ (𝐺 NeighbVtx 𝑎) ↔ {𝑐, 𝑎} ∈ 𝐸))
6159, 60bitrid 283 . . . . . . . . 9 (𝐺 ∈ USGraph → (𝑐𝑁 ↔ {𝑐, 𝑎} ∈ 𝐸))
6258, 61anbi12d 631 . . . . . . . 8 (𝐺 ∈ USGraph → ((𝑏𝑁𝑐𝑁) ↔ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)))
6362adantr 480 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝑎𝑉) → ((𝑏𝑁𝑐𝑁) ↔ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)))
64 tpex 7777 . . . . . . . . . 10 {𝑎, 𝑏, 𝑐} ∈ V
6564a1i 11 . . . . . . . . 9 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)) → {𝑎, 𝑏, 𝑐} ∈ V)
66 tpeq2 4768 . . . . . . . . . . . 12 (𝑦 = 𝑏 → {𝑎, 𝑦, 𝑧} = {𝑎, 𝑏, 𝑧})
6766eqeq2d 2745 . . . . . . . . . . 11 (𝑦 = 𝑏 → ({𝑎, 𝑏, 𝑐} = {𝑎, 𝑦, 𝑧} ↔ {𝑎, 𝑏, 𝑐} = {𝑎, 𝑏, 𝑧}))
68 preq2 4759 . . . . . . . . . . . . 13 (𝑦 = 𝑏 → {𝑎, 𝑦} = {𝑎, 𝑏})
6968eleq1d 2823 . . . . . . . . . . . 12 (𝑦 = 𝑏 → ({𝑎, 𝑦} ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
70 preq1 4758 . . . . . . . . . . . . 13 (𝑦 = 𝑏 → {𝑦, 𝑧} = {𝑏, 𝑧})
7170eleq1d 2823 . . . . . . . . . . . 12 (𝑦 = 𝑏 → ({𝑦, 𝑧} ∈ 𝐸 ↔ {𝑏, 𝑧} ∈ 𝐸))
7269, 713anbi13d 1438 . . . . . . . . . . 11 (𝑦 = 𝑏 → (({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸) ↔ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑏, 𝑧} ∈ 𝐸)))
7367, 723anbi13d 1438 . . . . . . . . . 10 (𝑦 = 𝑏 → (({𝑎, 𝑏, 𝑐} = {𝑎, 𝑦, 𝑧} ∧ (♯‘{𝑎, 𝑏, 𝑐}) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) ↔ ({𝑎, 𝑏, 𝑐} = {𝑎, 𝑏, 𝑧} ∧ (♯‘{𝑎, 𝑏, 𝑐}) = 3 ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑏, 𝑧} ∈ 𝐸))))
74 tpeq3 4769 . . . . . . . . . . . 12 (𝑧 = 𝑐 → {𝑎, 𝑏, 𝑧} = {𝑎, 𝑏, 𝑐})
7574eqeq2d 2745 . . . . . . . . . . 11 (𝑧 = 𝑐 → ({𝑎, 𝑏, 𝑐} = {𝑎, 𝑏, 𝑧} ↔ {𝑎, 𝑏, 𝑐} = {𝑎, 𝑏, 𝑐}))
76 preq2 4759 . . . . . . . . . . . . 13 (𝑧 = 𝑐 → {𝑎, 𝑧} = {𝑎, 𝑐})
7776eleq1d 2823 . . . . . . . . . . . 12 (𝑧 = 𝑐 → ({𝑎, 𝑧} ∈ 𝐸 ↔ {𝑎, 𝑐} ∈ 𝐸))
78 preq2 4759 . . . . . . . . . . . . 13 (𝑧 = 𝑐 → {𝑏, 𝑧} = {𝑏, 𝑐})
7978eleq1d 2823 . . . . . . . . . . . 12 (𝑧 = 𝑐 → ({𝑏, 𝑧} ∈ 𝐸 ↔ {𝑏, 𝑐} ∈ 𝐸))
8077, 793anbi23d 1439 . . . . . . . . . . 11 (𝑧 = 𝑐 → (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑏, 𝑧} ∈ 𝐸) ↔ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑎, 𝑐} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸)))
8175, 803anbi13d 1438 . . . . . . . . . 10 (𝑧 = 𝑐 → (({𝑎, 𝑏, 𝑐} = {𝑎, 𝑏, 𝑧} ∧ (♯‘{𝑎, 𝑏, 𝑐}) = 3 ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑏, 𝑧} ∈ 𝐸)) ↔ ({𝑎, 𝑏, 𝑐} = {𝑎, 𝑏, 𝑐} ∧ (♯‘{𝑎, 𝑏, 𝑐}) = 3 ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑎, 𝑐} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))))
82 usgruhgr 29212 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph)
8382adantr 480 . . . . . . . . . . . 12 ((𝐺 ∈ USGraph ∧ 𝑎𝑉) → 𝐺 ∈ UHGraph)
842eleq2i 2830 . . . . . . . . . . . . . 14 ({𝑏, 𝑎} ∈ 𝐸 ↔ {𝑏, 𝑎} ∈ (Edg‘𝐺))
8584biimpi 216 . . . . . . . . . . . . 13 ({𝑏, 𝑎} ∈ 𝐸 → {𝑏, 𝑎} ∈ (Edg‘𝐺))
8685adantr 480 . . . . . . . . . . . 12 (({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) → {𝑏, 𝑎} ∈ (Edg‘𝐺))
87 vex 3486 . . . . . . . . . . . . . 14 𝑏 ∈ V
8887prid1 4787 . . . . . . . . . . . . 13 𝑏 ∈ {𝑏, 𝑎}
8988a1i 11 . . . . . . . . . . . 12 ((𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸) → 𝑏 ∈ {𝑏, 𝑎})
90 uhgredgrnv 29156 . . . . . . . . . . . 12 ((𝐺 ∈ UHGraph ∧ {𝑏, 𝑎} ∈ (Edg‘𝐺) ∧ 𝑏 ∈ {𝑏, 𝑎}) → 𝑏 ∈ (Vtx‘𝐺))
9183, 86, 89, 90syl3an 1160 . . . . . . . . . . 11 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)) → 𝑏 ∈ (Vtx‘𝐺))
9291, 1eleqtrrdi 2849 . . . . . . . . . 10 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)) → 𝑏𝑉)
932eleq2i 2830 . . . . . . . . . . . . . 14 ({𝑐, 𝑎} ∈ 𝐸 ↔ {𝑐, 𝑎} ∈ (Edg‘𝐺))
9493biimpi 216 . . . . . . . . . . . . 13 ({𝑐, 𝑎} ∈ 𝐸 → {𝑐, 𝑎} ∈ (Edg‘𝐺))
9594adantl 481 . . . . . . . . . . . 12 (({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) → {𝑐, 𝑎} ∈ (Edg‘𝐺))
96 vex 3486 . . . . . . . . . . . . . 14 𝑐 ∈ V
9796prid1 4787 . . . . . . . . . . . . 13 𝑐 ∈ {𝑐, 𝑎}
9897a1i 11 . . . . . . . . . . . 12 ((𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸) → 𝑐 ∈ {𝑐, 𝑎})
99 uhgredgrnv 29156 . . . . . . . . . . . 12 ((𝐺 ∈ UHGraph ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺) ∧ 𝑐 ∈ {𝑐, 𝑎}) → 𝑐 ∈ (Vtx‘𝐺))
10083, 95, 98, 99syl3an 1160 . . . . . . . . . . 11 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)) → 𝑐 ∈ (Vtx‘𝐺))
101100, 1eleqtrrdi 2849 . . . . . . . . . 10 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)) → 𝑐𝑉)
102 eqidd 2735 . . . . . . . . . . 11 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)) → {𝑎, 𝑏, 𝑐} = {𝑎, 𝑏, 𝑐})
1032usgredgne 29232 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ USGraph ∧ {𝑏, 𝑎} ∈ 𝐸) → 𝑏𝑎)
104103necomd 2998 . . . . . . . . . . . . . . 15 ((𝐺 ∈ USGraph ∧ {𝑏, 𝑎} ∈ 𝐸) → 𝑎𝑏)
105104ad2ant2r 746 . . . . . . . . . . . . . 14 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)) → 𝑎𝑏)
1061053adant3 1132 . . . . . . . . . . . . 13 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)) → 𝑎𝑏)
107 simpl 482 . . . . . . . . . . . . . 14 ((𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸) → 𝑏𝑐)
1081073ad2ant3 1135 . . . . . . . . . . . . 13 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)) → 𝑏𝑐)
1092usgredgne 29232 . . . . . . . . . . . . . . 15 ((𝐺 ∈ USGraph ∧ {𝑐, 𝑎} ∈ 𝐸) → 𝑐𝑎)
110109ad2ant2rl 748 . . . . . . . . . . . . . 14 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)) → 𝑐𝑎)
1111103adant3 1132 . . . . . . . . . . . . 13 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)) → 𝑐𝑎)
112106, 108, 1113jca 1128 . . . . . . . . . . . 12 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)) → (𝑎𝑏𝑏𝑐𝑐𝑎))
113 hashtpg 14530 . . . . . . . . . . . . 13 ((𝑎 ∈ V ∧ 𝑏 ∈ V ∧ 𝑐 ∈ V) → ((𝑎𝑏𝑏𝑐𝑐𝑎) ↔ (♯‘{𝑎, 𝑏, 𝑐}) = 3))
114113el3v 3491 . . . . . . . . . . . 12 ((𝑎𝑏𝑏𝑐𝑐𝑎) ↔ (♯‘{𝑎, 𝑏, 𝑐}) = 3)
115112, 114sylib 218 . . . . . . . . . . 11 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)) → (♯‘{𝑎, 𝑏, 𝑐}) = 3)
116 prcom 4757 . . . . . . . . . . . . . . . 16 {𝑏, 𝑎} = {𝑎, 𝑏}
117116eleq1i 2829 . . . . . . . . . . . . . . 15 ({𝑏, 𝑎} ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸)
118117biimpi 216 . . . . . . . . . . . . . 14 ({𝑏, 𝑎} ∈ 𝐸 → {𝑎, 𝑏} ∈ 𝐸)
119118adantr 480 . . . . . . . . . . . . 13 (({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) → {𝑎, 𝑏} ∈ 𝐸)
1201193ad2ant2 1134 . . . . . . . . . . . 12 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)) → {𝑎, 𝑏} ∈ 𝐸)
121 prcom 4757 . . . . . . . . . . . . . . . 16 {𝑐, 𝑎} = {𝑎, 𝑐}
122121eleq1i 2829 . . . . . . . . . . . . . . 15 ({𝑐, 𝑎} ∈ 𝐸 ↔ {𝑎, 𝑐} ∈ 𝐸)
123122biimpi 216 . . . . . . . . . . . . . 14 ({𝑐, 𝑎} ∈ 𝐸 → {𝑎, 𝑐} ∈ 𝐸)
124123adantl 481 . . . . . . . . . . . . 13 (({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) → {𝑎, 𝑐} ∈ 𝐸)
1251243ad2ant2 1134 . . . . . . . . . . . 12 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)) → {𝑎, 𝑐} ∈ 𝐸)
126 simpr 484 . . . . . . . . . . . . 13 ((𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸) → {𝑏, 𝑐} ∈ 𝐸)
1271263ad2ant3 1135 . . . . . . . . . . . 12 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)) → {𝑏, 𝑐} ∈ 𝐸)
128120, 125, 1273jca 1128 . . . . . . . . . . 11 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)) → ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑎, 𝑐} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))
129102, 115, 1283jca 1128 . . . . . . . . . 10 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)) → ({𝑎, 𝑏, 𝑐} = {𝑎, 𝑏, 𝑐} ∧ (♯‘{𝑎, 𝑏, 𝑐}) = 3 ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑎, 𝑐} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸)))
13073, 81, 92, 101, 1292rspcedvdw 3645 . . . . . . . . 9 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)) → ∃𝑦𝑉𝑧𝑉 ({𝑎, 𝑏, 𝑐} = {𝑎, 𝑦, 𝑧} ∧ (♯‘{𝑎, 𝑏, 𝑐}) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))
131 eqeq1 2738 . . . . . . . . . . 11 (𝑡 = {𝑎, 𝑏, 𝑐} → (𝑡 = {𝑎, 𝑦, 𝑧} ↔ {𝑎, 𝑏, 𝑐} = {𝑎, 𝑦, 𝑧}))
132 fveqeq2 6928 . . . . . . . . . . 11 (𝑡 = {𝑎, 𝑏, 𝑐} → ((♯‘𝑡) = 3 ↔ (♯‘{𝑎, 𝑏, 𝑐}) = 3))
133131, 1323anbi12d 1437 . . . . . . . . . 10 (𝑡 = {𝑎, 𝑏, 𝑐} → ((𝑡 = {𝑎, 𝑦, 𝑧} ∧ (♯‘𝑡) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) ↔ ({𝑎, 𝑏, 𝑐} = {𝑎, 𝑦, 𝑧} ∧ (♯‘{𝑎, 𝑏, 𝑐}) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))
1341332rexbidv 3223 . . . . . . . . 9 (𝑡 = {𝑎, 𝑏, 𝑐} → (∃𝑦𝑉𝑧𝑉 (𝑡 = {𝑎, 𝑦, 𝑧} ∧ (♯‘𝑡) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) ↔ ∃𝑦𝑉𝑧𝑉 ({𝑎, 𝑏, 𝑐} = {𝑎, 𝑦, 𝑧} ∧ (♯‘{𝑎, 𝑏, 𝑐}) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))
13565, 130, 134spcedv 3607 . . . . . . . 8 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)) → ∃𝑡𝑦𝑉𝑧𝑉 (𝑡 = {𝑎, 𝑦, 𝑧} ∧ (♯‘𝑡) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))
1361353exp 1119 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝑎𝑉) → (({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) → ((𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸) → ∃𝑡𝑦𝑉𝑧𝑉 (𝑡 = {𝑎, 𝑦, 𝑧} ∧ (♯‘𝑡) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))))
13763, 136sylbid 240 . . . . . 6 ((𝐺 ∈ USGraph ∧ 𝑎𝑉) → ((𝑏𝑁𝑐𝑁) → ((𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸) → ∃𝑡𝑦𝑉𝑧𝑉 (𝑡 = {𝑎, 𝑦, 𝑧} ∧ (♯‘𝑡) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))))
138137rexlimdvv 3214 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑎𝑉) → (∃𝑏𝑁𝑐𝑁 (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸) → ∃𝑡𝑦𝑉𝑧𝑉 (𝑡 = {𝑎, 𝑦, 𝑧} ∧ (♯‘𝑡) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))
13955, 138impbid 212 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑎𝑉) → (∃𝑡𝑦𝑉𝑧𝑉 (𝑡 = {𝑎, 𝑦, 𝑧} ∧ (♯‘𝑡) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) ↔ ∃𝑏𝑁𝑐𝑁 (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)))
140139rexbidva 3179 . . 3 (𝐺 ∈ USGraph → (∃𝑎𝑉𝑡𝑦𝑉𝑧𝑉 (𝑡 = {𝑎, 𝑦, 𝑧} ∧ (♯‘𝑡) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) ↔ ∃𝑎𝑉𝑏𝑁𝑐𝑁 (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)))
1415, 140bitr3id 285 . 2 (𝐺 ∈ USGraph → (∃𝑡𝑎𝑉𝑦𝑉𝑧𝑉 (𝑡 = {𝑎, 𝑦, 𝑧} ∧ (♯‘𝑡) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) ↔ ∃𝑎𝑉𝑏𝑁𝑐𝑁 (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)))
1424, 141bitrid 283 1 (𝐺 ∈ USGraph → (∃𝑡 𝑡 ∈ (GrTriangles‘𝐺) ↔ ∃𝑎𝑉𝑏𝑁𝑐𝑁 (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1537  wex 1777  wcel 2103  wne 2942  wrex 3072  Vcvv 3482  {cpr 4650  {ctp 4652  cfv 6572  (class class class)co 7445  3c3 12345  chash 14375  Vtxcvtx 29022  Edgcedg 29073  UHGraphcuhgr 29082  USGraphcusgr 29175   NeighbVtx cnbgr 29358  GrTrianglescgrtri 47718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766  ax-cnex 11236  ax-resscn 11237  ax-1cn 11238  ax-icn 11239  ax-addcl 11240  ax-addrcl 11241  ax-mulcl 11242  ax-mulrcl 11243  ax-mulcom 11244  ax-addass 11245  ax-mulass 11246  ax-distr 11247  ax-i2m1 11248  ax-1ne0 11249  ax-1rid 11250  ax-rnegex 11251  ax-rrecex 11252  ax-cnre 11253  ax-pre-lttri 11254  ax-pre-lttrn 11255  ax-pre-ltadd 11256  ax-pre-mulgt0 11257
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4973  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-om 7900  df-1st 8026  df-2nd 8027  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-rdg 8462  df-1o 8518  df-2o 8519  df-3o 8520  df-oadd 8522  df-er 8759  df-en 9000  df-dom 9001  df-sdom 9002  df-fin 9003  df-dju 9966  df-card 10004  df-pnf 11322  df-mnf 11323  df-xr 11324  df-ltxr 11325  df-le 11326  df-sub 11518  df-neg 11519  df-nn 12290  df-2 12352  df-3 12353  df-n0 12550  df-xnn0 12622  df-z 12636  df-uz 12900  df-fz 13564  df-fzo 13708  df-hash 14376  df-edg 29074  df-uhgr 29084  df-upgr 29108  df-umgr 29109  df-uspgr 29176  df-usgr 29177  df-nbgr 29359  df-grtri 47719
This theorem is referenced by:  usgrexmpl2trifr  47772
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