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Theorem cplgr3v 26312
 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 2629 . . . 4 {𝐴, 𝐵, 𝐶} = (Vtx‘𝐺)
32iscplgrnb 26293 . . 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 4178 . . . . . 6 (𝑣 = 𝐴 → {𝑣} = {𝐴})
65difeq2d 3720 . . . . 5 (𝑣 = 𝐴 → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐴}))
7 tprot 4275 . . . . . . . 8 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
87difeq1i 3716 . . . . . . 7 ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) = ({𝐵, 𝐶, 𝐴} ∖ {𝐴})
9 necom 2844 . . . . . . . . 9 (𝐴𝐵𝐵𝐴)
10 necom 2844 . . . . . . . . 9 (𝐴𝐶𝐶𝐴)
11 diftpsn3 4323 . . . . . . . . 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 2666 . . . . . 6 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) = {𝐵, 𝐶})
15143ad2ant3 1082 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) = {𝐵, 𝐶})
166, 15sylan9eqr 2676 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ 𝑣 = 𝐴) → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = {𝐵, 𝐶})
17 oveq2 6643 . . . . . 6 (𝑣 = 𝐴 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝐴))
1817eleq2d 2685 . . . . 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 3147 . . 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 4178 . . . . . 6 (𝑣 = 𝐵 → {𝑣} = {𝐵})
2221difeq2d 3720 . . . . 5 (𝑣 = 𝐵 → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐵}))
23 tprot 4275 . . . . . . . . 9 {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
2423eqcomi 2629 . . . . . . . 8 {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴, 𝐵}
2524difeq1i 3716 . . . . . . 7 ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) = ({𝐶, 𝐴, 𝐵} ∖ {𝐵})
26 necom 2844 . . . . . . . . . . . 12 (𝐵𝐶𝐶𝐵)
2726biimpi 206 . . . . . . . . . . 11 (𝐵𝐶𝐶𝐵)
2827anim2i 592 . . . . . . . . . 10 ((𝐴𝐵𝐵𝐶) → (𝐴𝐵𝐶𝐵))
2928ancomd 467 . . . . . . . . 9 ((𝐴𝐵𝐵𝐶) → (𝐶𝐵𝐴𝐵))
30 diftpsn3 4323 . . . . . . . . 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 2666 . . . . . 6 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) = {𝐶, 𝐴})
34333ad2ant3 1082 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) = {𝐶, 𝐴})
3522, 34sylan9eqr 2676 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ 𝑣 = 𝐵) → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = {𝐶, 𝐴})
36 oveq2 6643 . . . . . 6 (𝑣 = 𝐵 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝐵))
3736eleq2d 2685 . . . . 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 3147 . . 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 4178 . . . . . 6 (𝑣 = 𝐶 → {𝑣} = {𝐶})
4140difeq2d 3720 . . . . 5 (𝑣 = 𝐶 → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐶}))
42 diftpsn3 4323 . . . . . . 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 2676 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ 𝑣 = 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = {𝐴, 𝐵})
46 oveq2 6643 . . . . . 6 (𝑣 = 𝐶 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝐶))
4746eleq2d 2685 . . . . 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 3147 . . 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 4306 . 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 2687 . . . . . . 7 (𝑛 = 𝐵 → (𝑛 ∈ (𝐺 NeighbVtx 𝐴) ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
58 eleq1 2687 . . . . . . 7 (𝑛 = 𝐶 → (𝑛 ∈ (𝐺 NeighbVtx 𝐴) ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
5957, 58ralprg 4225 . . . . . 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 2687 . . . . . . . 8 (𝑛 = 𝐶 → (𝑛 ∈ (𝐺 NeighbVtx 𝐵) ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐵)))
62 eleq1 2687 . . . . . . . 8 (𝑛 = 𝐴 → (𝑛 ∈ (𝐺 NeighbVtx 𝐵) ↔ 𝐴 ∈ (𝐺 NeighbVtx 𝐵)))
6361, 62ralprg 4225 . . . . . . 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 2687 . . . . . . 7 (𝑛 = 𝐴 → (𝑛 ∈ (𝐺 NeighbVtx 𝐶) ↔ 𝐴 ∈ (𝐺 NeighbVtx 𝐶)))
67 eleq1 2687 . . . . . . 7 (𝑛 = 𝐵 → (𝑛 ∈ (𝐺 NeighbVtx 𝐶) ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)))
6866, 67ralprg 4225 . . . . . 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 1397 . . . 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 1407 . . . 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 26246 . . . . . . 7 (𝐺 ∈ UPGraph → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ 𝐴 ∈ (𝐺 NeighbVtx 𝐵)))
75 nbgrsym 26246 . . . . . . 7 (𝐺 ∈ UPGraph → (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)))
76 nbgrsym 26246 . . . . . . 7 (𝐺 ∈ UPGraph → (𝐴 ∈ (𝐺 NeighbVtx 𝐶) ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
7774, 75, 763anbi123d 1397 . . . . . 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 4296 . . . . . . . . 9 (𝐴𝑋𝐴 ∈ {𝐵, 𝐶, 𝐴})
8786, 7syl6eleqr 2710 . . . . . . . 8 (𝐴𝑋𝐴 ∈ {𝐴, 𝐵, 𝐶})
88 tpid3g 4296 . . . . . . . . 9 (𝐵𝑌𝐵 ∈ {𝐶, 𝐴, 𝐵})
8988, 24syl6eleqr 2710 . . . . . . . 8 (𝐵𝑌𝐵 ∈ {𝐴, 𝐵, 𝐶})
90 tpid3g 4296 . . . . . . . 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 26222 . . . . . . . 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 26222 . . . . . . . 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 2846 . . . . . . . . . 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 26222 . . . . . . . 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 1397 . . . . . 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 1357 . . . . 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 1481   ∈ wcel 1988   ≠ wne 2791  ∀wral 2909   ∖ cdif 3564  {csn 4168  {cpr 4170  {ctp 4172  ‘cfv 5876  (class class class)co 6635  Vtxcvtx 25855  Edgcedg 25920   UPGraph cupgr 25956   NeighbVtx cnbgr 26205  ComplGraphccplgr 26207 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-card 8750  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-n0 11278  df-xnn0 11349  df-z 11363  df-uz 11673  df-fz 12312  df-hash 13101  df-edg 25921  df-upgr 25958  df-nbgr 26209  df-uvtxa 26211  df-cplgr 26212 This theorem is referenced by:  cusgr3vnbpr  26313
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