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Theorem usgrgrtrirex 47801
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 47796 . . 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 3294 . . 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 6931 . . . . . . . . . . 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 3009 . . . . . . . . . . . . . 14 (𝑏 = 𝑦 → (𝑏𝑐𝑦𝑐))
9 preq1 4758 . . . . . . . . . . . . . . 15 (𝑏 = 𝑦 → {𝑏, 𝑐} = {𝑦, 𝑐})
109eleq1d 2829 . . . . . . . . . . . . . 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 3010 . . . . . . . . . . . . . 14 (𝑐 = 𝑧 → (𝑦𝑐𝑦𝑧))
13 preq2 4759 . . . . . . . . . . . . . . 15 (𝑐 = 𝑧 → {𝑦, 𝑐} = {𝑦, 𝑧})
1413eleq1d 2829 . . . . . . . . . . . . . 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 2835 . . . . . . . . . . . . . . . . . . . . 21 ({𝑎, 𝑦} ∈ 𝐸 ↔ {𝑦, 𝑎} ∈ 𝐸)
182nbusgreledg 29390 . . . . . . . . . . . . . . . . . . . . . 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 2855 . . . . . . . . . . . . 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 2835 . . . . . . . . . . . . . . . . . . . . 21 ({𝑎, 𝑧} ∈ 𝐸 ↔ {𝑧, 𝑎} ∈ 𝐸)
312nbusgreledg 29390 . . . . . . . . . . . . . . . . . . . . . 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 2855 . . . . . . . . . . . . 13 ((((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ (𝑦𝑉𝑧𝑉)) ∧ (♯‘{𝑎, 𝑦, 𝑧}) = 3 ∧ ({𝑎, 𝑦} ∈ 𝐸 ∧ {𝑎, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → 𝑧𝑁)
41 hashtpg 14536 . . . . . . . . . . . . . . . . . 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 3496 . . . . . . . . . . . . . . . 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 3649 . . . . . . . . . . . 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 3219 . . . . . 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 2836 . . . . . . . . . 10 (𝑏𝑁𝑏 ∈ (𝐺 NeighbVtx 𝑎))
572nbusgreledg 29390 . . . . . . . . . 10 (𝐺 ∈ USGraph → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ {𝑏, 𝑎} ∈ 𝐸))
5856, 57bitrid 283 . . . . . . . . 9 (𝐺 ∈ USGraph → (𝑏𝑁 ↔ {𝑏, 𝑎} ∈ 𝐸))
5927eleq2i 2836 . . . . . . . . . 10 (𝑐𝑁𝑐 ∈ (𝐺 NeighbVtx 𝑎))
602nbusgreledg 29390 . . . . . . . . . 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 7783 . . . . . . . . . 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 2751 . . . . . . . . . . 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 2829 . . . . . . . . . . . 12 (𝑦 = 𝑏 → ({𝑎, 𝑦} ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
70 preq1 4758 . . . . . . . . . . . . 13 (𝑦 = 𝑏 → {𝑦, 𝑧} = {𝑏, 𝑧})
7170eleq1d 2829 . . . . . . . . . . . 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 2751 . . . . . . . . . . 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 2829 . . . . . . . . . . . 12 (𝑧 = 𝑐 → ({𝑎, 𝑧} ∈ 𝐸 ↔ {𝑎, 𝑐} ∈ 𝐸))
78 preq2 4759 . . . . . . . . . . . . 13 (𝑧 = 𝑐 → {𝑏, 𝑧} = {𝑏, 𝑐})
7978eleq1d 2829 . . . . . . . . . . . 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 29223 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph)
8382adantr 480 . . . . . . . . . . . 12 ((𝐺 ∈ USGraph ∧ 𝑎𝑉) → 𝐺 ∈ UHGraph)
842eleq2i 2836 . . . . . . . . . . . . . 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 3492 . . . . . . . . . . . . . 14 𝑏 ∈ V
8887prid1 4787 . . . . . . . . . . . . 13 𝑏 ∈ {𝑏, 𝑎}
8988a1i 11 . . . . . . . . . . . 12 ((𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸) → 𝑏 ∈ {𝑏, 𝑎})
90 uhgredgrnv 29167 . . . . . . . . . . . 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 2855 . . . . . . . . . 10 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)) → 𝑏𝑉)
932eleq2i 2836 . . . . . . . . . . . . . 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 3492 . . . . . . . . . . . . . 14 𝑐 ∈ V
9796prid1 4787 . . . . . . . . . . . . 13 𝑐 ∈ {𝑐, 𝑎}
9897a1i 11 . . . . . . . . . . . 12 ((𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸) → 𝑐 ∈ {𝑐, 𝑎})
99 uhgredgrnv 29167 . . . . . . . . . . . 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 2855 . . . . . . . . . 10 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)) → 𝑐𝑉)
102 eqidd 2741 . . . . . . . . . . 11 (((𝐺 ∈ USGraph ∧ 𝑎𝑉) ∧ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)) → {𝑎, 𝑏, 𝑐} = {𝑎, 𝑏, 𝑐})
1032usgredgne 29243 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ USGraph ∧ {𝑏, 𝑎} ∈ 𝐸) → 𝑏𝑎)
104103necomd 3002 . . . . . . . . . . . . . . 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 29243 . . . . . . . . . . . . . . 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 14536 . . . . . . . . . . . . 13 ((𝑎 ∈ V ∧ 𝑏 ∈ V ∧ 𝑐 ∈ V) → ((𝑎𝑏𝑏𝑐𝑐𝑎) ↔ (♯‘{𝑎, 𝑏, 𝑐}) = 3))
114113el3v 3496 . . . . . . . . . . . 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 2835 . . . . . . . . . . . . . . 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 2835 . . . . . . . . . . . . . . 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 3649 . . . . . . . . 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 2744 . . . . . . . . . . 11 (𝑡 = {𝑎, 𝑏, 𝑐} → (𝑡 = {𝑎, 𝑦, 𝑧} ↔ {𝑎, 𝑏, 𝑐} = {𝑎, 𝑦, 𝑧}))
132 fveqeq2 6931 . . . . . . . . . . 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 3228 . . . . . . . . 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 3611 . . . . . . . 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 3218 . . . . 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 3183 . . 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 2108  wne 2946  wrex 3076  Vcvv 3488  {cpr 4650  {ctp 4652  cfv 6575  (class class class)co 7450  3c3 12351  chash 14381  Vtxcvtx 29033  Edgcedg 29084  UHGraphcuhgr 29093  USGraphcusgr 29186   NeighbVtx cnbgr 29369  GrTrianglescgrtri 47790
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 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772  ax-cnex 11242  ax-resscn 11243  ax-1cn 11244  ax-icn 11245  ax-addcl 11246  ax-addrcl 11247  ax-mulcl 11248  ax-mulrcl 11249  ax-mulcom 11250  ax-addass 11251  ax-mulass 11252  ax-distr 11253  ax-i2m1 11254  ax-1ne0 11255  ax-1rid 11256  ax-rnegex 11257  ax-rrecex 11258  ax-cnre 11259  ax-pre-lttri 11260  ax-pre-lttrn 11261  ax-pre-ltadd 11262  ax-pre-mulgt0 11263
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 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6334  df-ord 6400  df-on 6401  df-lim 6402  df-suc 6403  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-riota 7406  df-ov 7453  df-oprab 7454  df-mpo 7455  df-om 7906  df-1st 8032  df-2nd 8033  df-frecs 8324  df-wrecs 8355  df-recs 8429  df-rdg 8468  df-1o 8524  df-2o 8525  df-3o 8526  df-oadd 8528  df-er 8765  df-en 9006  df-dom 9007  df-sdom 9008  df-fin 9009  df-dju 9972  df-card 10010  df-pnf 11328  df-mnf 11329  df-xr 11330  df-ltxr 11331  df-le 11332  df-sub 11524  df-neg 11525  df-nn 12296  df-2 12358  df-3 12359  df-n0 12556  df-xnn0 12628  df-z 12642  df-uz 12906  df-fz 13570  df-fzo 13714  df-hash 14382  df-edg 29085  df-uhgr 29095  df-upgr 29119  df-umgr 29120  df-uspgr 29187  df-usgr 29188  df-nbgr 29370  df-grtri 47791
This theorem is referenced by:  usgrexmpl2trifr  47854
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