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Theorem cplgr3v 26235
Description: A pseudograph with three (different) vertices is complete iff there is an edge between each of these three vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 5-Nov-2020.)
Hypotheses
Ref Expression
cplgr3v.e 𝐸 = (Edg‘𝐺)
cplgr3v.t (Vtx‘𝐺) = {𝐴, 𝐵, 𝐶}
Assertion
Ref Expression
cplgr3v (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐺 ∈ ComplGraph ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))

Proof of Theorem cplgr3v
Dummy variables 𝑛 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cplgr3v.t . . . . 5 (Vtx‘𝐺) = {𝐴, 𝐵, 𝐶}
21eqcomi 2630 . . . 4 {𝐴, 𝐵, 𝐶} = (Vtx‘𝐺)
32iscplgrnb 26216 . . 3 (𝐺 ∈ UPGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ {𝐴, 𝐵, 𝐶}∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
433ad2ant2 1081 . 2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ {𝐴, 𝐵, 𝐶}∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
5 sneq 4163 . . . . . 6 (𝑣 = 𝐴 → {𝑣} = {𝐴})
65difeq2d 3711 . . . . 5 (𝑣 = 𝐴 → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐴}))
7 tprot 4259 . . . . . . . 8 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
87difeq1i 3707 . . . . . . 7 ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) = ({𝐵, 𝐶, 𝐴} ∖ {𝐴})
9 necom 2843 . . . . . . . . 9 (𝐴𝐵𝐵𝐴)
10 necom 2843 . . . . . . . . 9 (𝐴𝐶𝐶𝐴)
11 diftpsn3 4306 . . . . . . . . 9 ((𝐵𝐴𝐶𝐴) → ({𝐵, 𝐶, 𝐴} ∖ {𝐴}) = {𝐵, 𝐶})
129, 10, 11syl2anb 496 . . . . . . . 8 ((𝐴𝐵𝐴𝐶) → ({𝐵, 𝐶, 𝐴} ∖ {𝐴}) = {𝐵, 𝐶})
13123adant3 1079 . . . . . . 7 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐵, 𝐶, 𝐴} ∖ {𝐴}) = {𝐵, 𝐶})
148, 13syl5eq 2667 . . . . . 6 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) = {𝐵, 𝐶})
15143ad2ant3 1082 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) = {𝐵, 𝐶})
166, 15sylan9eqr 2677 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ 𝑣 = 𝐴) → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = {𝐵, 𝐶})
17 oveq2 6618 . . . . . 6 (𝑣 = 𝐴 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝐴))
1817eleq2d 2684 . . . . 5 (𝑣 = 𝐴 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝐴)))
1918adantl 482 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ 𝑣 = 𝐴) → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝐴)))
2016, 19raleqbidv 3144 . . 3 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ 𝑣 = 𝐴) → (∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ {𝐵, 𝐶}𝑛 ∈ (𝐺 NeighbVtx 𝐴)))
21 sneq 4163 . . . . . 6 (𝑣 = 𝐵 → {𝑣} = {𝐵})
2221difeq2d 3711 . . . . 5 (𝑣 = 𝐵 → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐵}))
23 tprot 4259 . . . . . . . . 9 {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
2423eqcomi 2630 . . . . . . . 8 {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴, 𝐵}
2524difeq1i 3707 . . . . . . 7 ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) = ({𝐶, 𝐴, 𝐵} ∖ {𝐵})
26 necom 2843 . . . . . . . . . . . 12 (𝐵𝐶𝐶𝐵)
2726biimpi 206 . . . . . . . . . . 11 (𝐵𝐶𝐶𝐵)
2827anim2i 592 . . . . . . . . . 10 ((𝐴𝐵𝐵𝐶) → (𝐴𝐵𝐶𝐵))
2928ancomd 467 . . . . . . . . 9 ((𝐴𝐵𝐵𝐶) → (𝐶𝐵𝐴𝐵))
30 diftpsn3 4306 . . . . . . . . 9 ((𝐶𝐵𝐴𝐵) → ({𝐶, 𝐴, 𝐵} ∖ {𝐵}) = {𝐶, 𝐴})
3129, 30syl 17 . . . . . . . 8 ((𝐴𝐵𝐵𝐶) → ({𝐶, 𝐴, 𝐵} ∖ {𝐵}) = {𝐶, 𝐴})
32313adant2 1078 . . . . . . 7 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐶, 𝐴, 𝐵} ∖ {𝐵}) = {𝐶, 𝐴})
3325, 32syl5eq 2667 . . . . . 6 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) = {𝐶, 𝐴})
34333ad2ant3 1082 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) = {𝐶, 𝐴})
3522, 34sylan9eqr 2677 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ 𝑣 = 𝐵) → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = {𝐶, 𝐴})
36 oveq2 6618 . . . . . 6 (𝑣 = 𝐵 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝐵))
3736eleq2d 2684 . . . . 5 (𝑣 = 𝐵 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝐵)))
3837adantl 482 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ 𝑣 = 𝐵) → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝐵)))
3935, 38raleqbidv 3144 . . 3 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ 𝑣 = 𝐵) → (∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵)))
40 sneq 4163 . . . . . 6 (𝑣 = 𝐶 → {𝑣} = {𝐶})
4140difeq2d 3711 . . . . 5 (𝑣 = 𝐶 → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐶}))
42 diftpsn3 4306 . . . . . . 7 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
43423adant1 1077 . . . . . 6 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
44433ad2ant3 1082 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
4541, 44sylan9eqr 2677 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ 𝑣 = 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = {𝐴, 𝐵})
46 oveq2 6618 . . . . . 6 (𝑣 = 𝐶 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝐶))
4746eleq2d 2684 . . . . 5 (𝑣 = 𝐶 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝐶)))
4847adantl 482 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ 𝑣 = 𝐶) → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝐶)))
4945, 48raleqbidv 3144 . . 3 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ 𝑣 = 𝐶) → (∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ {𝐴, 𝐵}𝑛 ∈ (𝐺 NeighbVtx 𝐶)))
50 simp1 1059 . . . 4 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐴𝑋)
51503ad2ant1 1080 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → 𝐴𝑋)
52 simp2 1060 . . . 4 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐵𝑌)
53523ad2ant1 1080 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → 𝐵𝑌)
54 simp3 1061 . . . 4 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶𝑍)
55543ad2ant1 1080 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → 𝐶𝑍)
5620, 39, 49, 51, 53, 55raltpd 4290 . 2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (∀𝑣 ∈ {𝐴, 𝐵, 𝐶}∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (∀𝑛 ∈ {𝐵, 𝐶}𝑛 ∈ (𝐺 NeighbVtx 𝐴) ∧ ∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵) ∧ ∀𝑛 ∈ {𝐴, 𝐵}𝑛 ∈ (𝐺 NeighbVtx 𝐶))))
57 eleq1 2686 . . . . . . 7 (𝑛 = 𝐵 → (𝑛 ∈ (𝐺 NeighbVtx 𝐴) ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
58 eleq1 2686 . . . . . . 7 (𝑛 = 𝐶 → (𝑛 ∈ (𝐺 NeighbVtx 𝐴) ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
5957, 58ralprg 4210 . . . . . 6 ((𝐵𝑌𝐶𝑍) → (∀𝑛 ∈ {𝐵, 𝐶}𝑛 ∈ (𝐺 NeighbVtx 𝐴) ↔ (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴))))
60593adant1 1077 . . . . 5 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∀𝑛 ∈ {𝐵, 𝐶}𝑛 ∈ (𝐺 NeighbVtx 𝐴) ↔ (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴))))
61 eleq1 2686 . . . . . . . 8 (𝑛 = 𝐶 → (𝑛 ∈ (𝐺 NeighbVtx 𝐵) ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐵)))
62 eleq1 2686 . . . . . . . 8 (𝑛 = 𝐴 → (𝑛 ∈ (𝐺 NeighbVtx 𝐵) ↔ 𝐴 ∈ (𝐺 NeighbVtx 𝐵)))
6361, 62ralprg 4210 . . . . . . 7 ((𝐶𝑍𝐴𝑋) → (∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵) ↔ (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵))))
6463ancoms 469 . . . . . 6 ((𝐴𝑋𝐶𝑍) → (∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵) ↔ (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵))))
65643adant2 1078 . . . . 5 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵) ↔ (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵))))
66 eleq1 2686 . . . . . . 7 (𝑛 = 𝐴 → (𝑛 ∈ (𝐺 NeighbVtx 𝐶) ↔ 𝐴 ∈ (𝐺 NeighbVtx 𝐶)))
67 eleq1 2686 . . . . . . 7 (𝑛 = 𝐵 → (𝑛 ∈ (𝐺 NeighbVtx 𝐶) ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)))
6866, 67ralprg 4210 . . . . . 6 ((𝐴𝑋𝐵𝑌) → (∀𝑛 ∈ {𝐴, 𝐵}𝑛 ∈ (𝐺 NeighbVtx 𝐶) ↔ (𝐴 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))))
69683adant3 1079 . . . . 5 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∀𝑛 ∈ {𝐴, 𝐵}𝑛 ∈ (𝐺 NeighbVtx 𝐶) ↔ (𝐴 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))))
7060, 65, 693anbi123d 1396 . . . 4 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((∀𝑛 ∈ {𝐵, 𝐶}𝑛 ∈ (𝐺 NeighbVtx 𝐴) ∧ ∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵) ∧ ∀𝑛 ∈ {𝐴, 𝐵}𝑛 ∈ (𝐺 NeighbVtx 𝐶)) ↔ ((𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵)) ∧ (𝐴 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)))))
71703ad2ant1 1080 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((∀𝑛 ∈ {𝐵, 𝐶}𝑛 ∈ (𝐺 NeighbVtx 𝐴) ∧ ∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵) ∧ ∀𝑛 ∈ {𝐴, 𝐵}𝑛 ∈ (𝐺 NeighbVtx 𝐶)) ↔ ((𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵)) ∧ (𝐴 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)))))
72 3an6 1406 . . . 4 (((𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵)) ∧ (𝐴 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))) ↔ ((𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐶)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))))
7372a1i 11 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (((𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵)) ∧ (𝐴 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))) ↔ ((𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐶)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)))))
74 nbgrsym 26169 . . . . . . 7 (𝐺 ∈ UPGraph → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ 𝐴 ∈ (𝐺 NeighbVtx 𝐵)))
75 nbgrsym 26169 . . . . . . 7 (𝐺 ∈ UPGraph → (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)))
76 nbgrsym 26169 . . . . . . 7 (𝐺 ∈ UPGraph → (𝐴 ∈ (𝐺 NeighbVtx 𝐶) ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
7774, 75, 763anbi123d 1396 . . . . . 6 (𝐺 ∈ UPGraph → ((𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐶)) ↔ (𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴))))
78773ad2ant2 1081 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐶)) ↔ (𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴))))
7978anbi1d 740 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (((𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐶)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))) ↔ ((𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)))))
80 3anrot 1041 . . . . . . . 8 ((𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)) ↔ (𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
8180bicomi 214 . . . . . . 7 ((𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ↔ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)))
8281anbi1i 730 . . . . . 6 (((𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))) ↔ ((𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))))
83 anidm 675 . . . . . 6 (((𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))) ↔ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)))
8482, 83bitri 264 . . . . 5 (((𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))) ↔ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)))
8584a1i 11 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (((𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))) ↔ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))))
86 tpid3g 4280 . . . . . . . . 9 (𝐴𝑋𝐴 ∈ {𝐵, 𝐶, 𝐴})
8786, 7syl6eleqr 2709 . . . . . . . 8 (𝐴𝑋𝐴 ∈ {𝐴, 𝐵, 𝐶})
88 tpid3g 4280 . . . . . . . . 9 (𝐵𝑌𝐵 ∈ {𝐶, 𝐴, 𝐵})
8988, 24syl6eleqr 2709 . . . . . . . 8 (𝐵𝑌𝐵 ∈ {𝐴, 𝐵, 𝐶})
90 tpid3g 4280 . . . . . . . 8 (𝐶𝑍𝐶 ∈ {𝐴, 𝐵, 𝐶})
9187, 89, 903anim123i 1245 . . . . . . 7 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
92 df-3an 1038 . . . . . . 7 ((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ↔ ((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
9391, 92sylib 208 . . . . . 6 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
94 simpr 477 . . . . . . . . . . . 12 ((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
9594adantr 481 . . . . . . . . . . 11 (((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
9695anim1i 591 . . . . . . . . . 10 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ) → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ UPGraph ))
9796ancomd 467 . . . . . . . . 9 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ) → (𝐺 ∈ UPGraph ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}))
98973adant3 1079 . . . . . . . 8 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐺 ∈ UPGraph ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}))
99 simpll 789 . . . . . . . . . 10 (((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) → 𝐴 ∈ {𝐴, 𝐵, 𝐶})
100 simp1 1059 . . . . . . . . . 10 ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝐴𝐵)
10199, 100anim12i 589 . . . . . . . . 9 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐴𝐵))
1021013adant2 1078 . . . . . . . 8 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐴𝐵))
103 cplgr3v.e . . . . . . . . 9 𝐸 = (Edg‘𝐺)
1042, 103nbupgrel 26145 . . . . . . . 8 (((𝐺 ∈ UPGraph ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐴𝐵)) → (𝐴 ∈ (𝐺 NeighbVtx 𝐵) ↔ {𝐴, 𝐵} ∈ 𝐸))
10598, 102, 104syl2anc 692 . . . . . . 7 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐴 ∈ (𝐺 NeighbVtx 𝐵) ↔ {𝐴, 𝐵} ∈ 𝐸))
106 simpr 477 . . . . . . . . . . 11 (((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
107106anim1i 591 . . . . . . . . . 10 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ) → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ UPGraph ))
108107ancomd 467 . . . . . . . . 9 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ) → (𝐺 ∈ UPGraph ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
1091083adant3 1079 . . . . . . . 8 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐺 ∈ UPGraph ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
110 simp3 1061 . . . . . . . . . 10 ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝐵𝐶)
11195, 110anim12i 589 . . . . . . . . 9 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵𝐶))
1121113adant2 1078 . . . . . . . 8 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵𝐶))
1132, 103nbupgrel 26145 . . . . . . . 8 (((𝐺 ∈ UPGraph ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ (𝐵 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵𝐶)) → (𝐵 ∈ (𝐺 NeighbVtx 𝐶) ↔ {𝐵, 𝐶} ∈ 𝐸))
114109, 112, 113syl2anc 692 . . . . . . 7 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐵 ∈ (𝐺 NeighbVtx 𝐶) ↔ {𝐵, 𝐶} ∈ 𝐸))
11599anim1i 591 . . . . . . . . . 10 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ) → (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ UPGraph ))
116115ancomd 467 . . . . . . . . 9 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ) → (𝐺 ∈ UPGraph ∧ 𝐴 ∈ {𝐴, 𝐵, 𝐶}))
1171163adant3 1079 . . . . . . . 8 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐺 ∈ UPGraph ∧ 𝐴 ∈ {𝐴, 𝐵, 𝐶}))
118 simp2 1060 . . . . . . . . . . 11 ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝐴𝐶)
119118necomd 2845 . . . . . . . . . 10 ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝐶𝐴)
120106, 119anim12i 589 . . . . . . . . 9 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶𝐴))
1211203adant2 1078 . . . . . . . 8 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶𝐴))
1222, 103nbupgrel 26145 . . . . . . . 8 (((𝐺 ∈ UPGraph ∧ 𝐴 ∈ {𝐴, 𝐵, 𝐶}) ∧ (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶𝐴)) → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐶, 𝐴} ∈ 𝐸))
123117, 121, 122syl2anc 692 . . . . . . 7 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐶, 𝐴} ∈ 𝐸))
124105, 114, 1233anbi123d 1396 . . . . . 6 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))
12593, 124syl3an1 1356 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))
12680, 125syl5bb 272 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))
12779, 85, 1263bitrd 294 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (((𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐶)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))
12871, 73, 1273bitrd 294 . 2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((∀𝑛 ∈ {𝐵, 𝐶}𝑛 ∈ (𝐺 NeighbVtx 𝐴) ∧ ∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵) ∧ ∀𝑛 ∈ {𝐴, 𝐵}𝑛 ∈ (𝐺 NeighbVtx 𝐶)) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))
1294, 56, 1283bitrd 294 1 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐺 ∈ ComplGraph ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  cdif 3556  {csn 4153  {cpr 4155  {ctp 4157  cfv 5852  (class class class)co 6610  Vtxcvtx 25791  Edgcedg 25856   UPGraph cupgr 25888   NeighbVtx cnbgr 26128  ComplGraphccplgr 26130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-card 8717  df-cda 8942  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-n0 11245  df-xnn0 11316  df-z 11330  df-uz 11640  df-fz 12277  df-hash 13066  df-edg 25857  df-upgr 25890  df-nbgr 26132  df-uvtxa 26134  df-cplgr 26135
This theorem is referenced by:  cusgr3vnbpr  26236
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