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Theorem cplgr3v 28383

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.) (Proof shortened by AV, 13-Feb-2022.)

**Hypotheses**
Ref	Expression
cplgr3v.e	⊢ 𝐸 = (Edg‘𝐺)
cplgr3v.t	⊢ (Vtx‘𝐺) = {𝐴, 𝐵, 𝐶}

**Assertion**
Ref	Expression
cplgr3v	⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (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.

Step	Hyp	Ref	Expression
1		cplgr3v.t	. . . . 5 ⊢ (Vtx‘𝐺) = {𝐴, 𝐵, 𝐶}
2	1	eqcomi 2745	. . . 4 ⊢ {𝐴, 𝐵, 𝐶} = (Vtx‘𝐺)
3	2	iscplgrnb 28364	. . 3 ⊢ (𝐺 ∈ UPGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ {𝐴, 𝐵, 𝐶}∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
4	3	3ad2ant2 1134	. 2 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ {𝐴, 𝐵, 𝐶}∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
5		sneq 4596	. . . . . 6 ⊢ (𝑣 = 𝐴 → {𝑣} = {𝐴})
6	5	difeq2d 4082	. . . . 5 ⊢ (𝑣 = 𝐴 → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐴}))
7		tprot 4710	. . . . . . . 8 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
8	7	difeq1i 4078	. . . . . . 7 ⊢ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) = ({𝐵, 𝐶, 𝐴} ∖ {𝐴})
9		necom 2997	. . . . . . . . 9 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
10		necom 2997	. . . . . . . . 9 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐶 ≠ 𝐴)
11		diftpsn3 4762	. . . . . . . . 9 ⊢ ((𝐵 ≠ 𝐴 ∧ 𝐶 ≠ 𝐴) → ({𝐵, 𝐶, 𝐴} ∖ {𝐴}) = {𝐵, 𝐶})
12	9, 10, 11	syl2anb 598	. . . . . . . 8 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ({𝐵, 𝐶, 𝐴} ∖ {𝐴}) = {𝐵, 𝐶})
13	12	3adant3 1132	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐵, 𝐶, 𝐴} ∖ {𝐴}) = {𝐵, 𝐶})
14	8, 13	eqtrid 2788	. . . . . 6 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) = {𝐵, 𝐶})
15	14	3ad2ant3 1135	. . . . 5 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) = {𝐵, 𝐶})
16	6, 15	sylan9eqr 2798	. . . 4 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ 𝑣 = 𝐴) → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = {𝐵, 𝐶})
17		oveq2 7365	. . . . . 6 ⊢ (𝑣 = 𝐴 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝐴))
18	17	eleq2d 2823	. . . . 5 ⊢ (𝑣 = 𝐴 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝐴)))
19	18	adantl 482	. . . 4 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ 𝑣 = 𝐴) → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝐴)))
20	16, 19	raleqbidv 3319	. . 3 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ 𝑣 = 𝐴) → (∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ {𝐵, 𝐶}𝑛 ∈ (𝐺 NeighbVtx 𝐴)))
21		sneq 4596	. . . . . 6 ⊢ (𝑣 = 𝐵 → {𝑣} = {𝐵})
22	21	difeq2d 4082	. . . . 5 ⊢ (𝑣 = 𝐵 → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐵}))
23		tprot 4710	. . . . . . . . 9 ⊢ {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
24	23	eqcomi 2745	. . . . . . . 8 ⊢ {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴, 𝐵}
25	24	difeq1i 4078	. . . . . . 7 ⊢ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) = ({𝐶, 𝐴, 𝐵} ∖ {𝐵})
26		necom 2997	. . . . . . . . . . . 12 ⊢ (𝐵 ≠ 𝐶 ↔ 𝐶 ≠ 𝐵)
27	26	biimpi 215	. . . . . . . . . . 11 ⊢ (𝐵 ≠ 𝐶 → 𝐶 ≠ 𝐵)
28	27	anim2i 617	. . . . . . . . . 10 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) → (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵))
29	28	ancomd 462	. . . . . . . . 9 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) → (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵))
30		diftpsn3 4762	. . . . . . . . 9 ⊢ ((𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵) → ({𝐶, 𝐴, 𝐵} ∖ {𝐵}) = {𝐶, 𝐴})
31	29, 30	syl 17	. . . . . . . 8 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) → ({𝐶, 𝐴, 𝐵} ∖ {𝐵}) = {𝐶, 𝐴})
32	31	3adant2 1131	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐶, 𝐴, 𝐵} ∖ {𝐵}) = {𝐶, 𝐴})
33	25, 32	eqtrid 2788	. . . . . 6 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) = {𝐶, 𝐴})
34	33	3ad2ant3 1135	. . . . 5 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) = {𝐶, 𝐴})
35	22, 34	sylan9eqr 2798	. . . 4 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ 𝑣 = 𝐵) → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = {𝐶, 𝐴})
36		oveq2 7365	. . . . . 6 ⊢ (𝑣 = 𝐵 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝐵))
37	36	eleq2d 2823	. . . . 5 ⊢ (𝑣 = 𝐵 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝐵)))
38	37	adantl 482	. . . 4 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ 𝑣 = 𝐵) → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝐵)))
39	35, 38	raleqbidv 3319	. . 3 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ 𝑣 = 𝐵) → (∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵)))
40		sneq 4596	. . . . . 6 ⊢ (𝑣 = 𝐶 → {𝑣} = {𝐶})
41	40	difeq2d 4082	. . . . 5 ⊢ (𝑣 = 𝐶 → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐶}))
42		diftpsn3 4762	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
43	42	3adant1 1130	. . . . . 6 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
44	43	3ad2ant3 1135	. . . . 5 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
45	41, 44	sylan9eqr 2798	. . . 4 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ 𝑣 = 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = {𝐴, 𝐵})
46		oveq2 7365	. . . . . 6 ⊢ (𝑣 = 𝐶 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝐶))
47	46	eleq2d 2823	. . . . 5 ⊢ (𝑣 = 𝐶 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝐶)))
48	47	adantl 482	. . . 4 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ 𝑣 = 𝐶) → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝐶)))
49	45, 48	raleqbidv 3319	. . 3 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ 𝑣 = 𝐶) → (∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ {𝐴, 𝐵}𝑛 ∈ (𝐺 NeighbVtx 𝐶)))
50		simp1 1136	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐴 ∈ 𝑋)
51	50	3ad2ant1 1133	. . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → 𝐴 ∈ 𝑋)
52		simp2 1137	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐵 ∈ 𝑌)
53	52	3ad2ant1 1133	. . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → 𝐵 ∈ 𝑌)
54		simp3 1138	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 ∈ 𝑍)
55	54	3ad2ant1 1133	. . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → 𝐶 ∈ 𝑍)
56	20, 39, 49, 51, 53, 55	raltpd 4742	. 2 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (∀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 2825	. . . . . . 7 ⊢ (𝑛 = 𝐵 → (𝑛 ∈ (𝐺 NeighbVtx 𝐴) ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
58		eleq1 2825	. . . . . . 7 ⊢ (𝑛 = 𝐶 → (𝑛 ∈ (𝐺 NeighbVtx 𝐴) ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
59	57, 58	ralprg 4655	. . . . . 6 ⊢ ((𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (∀𝑛 ∈ {𝐵, 𝐶}𝑛 ∈ (𝐺 NeighbVtx 𝐴) ↔ (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴))))
60	59	3adant1 1130	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (∀𝑛 ∈ {𝐵, 𝐶}𝑛 ∈ (𝐺 NeighbVtx 𝐴) ↔ (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴))))
61		eleq1 2825	. . . . . . . 8 ⊢ (𝑛 = 𝐶 → (𝑛 ∈ (𝐺 NeighbVtx 𝐵) ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐵)))
62		eleq1 2825	. . . . . . . 8 ⊢ (𝑛 = 𝐴 → (𝑛 ∈ (𝐺 NeighbVtx 𝐵) ↔ 𝐴 ∈ (𝐺 NeighbVtx 𝐵)))
63	61, 62	ralprg 4655	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋) → (∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵) ↔ (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵))))
64	63	ancoms 459	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑍) → (∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵) ↔ (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵))))
65	64	3adant2 1131	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵) ↔ (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵))))
66		eleq1 2825	. . . . . . 7 ⊢ (𝑛 = 𝐴 → (𝑛 ∈ (𝐺 NeighbVtx 𝐶) ↔ 𝐴 ∈ (𝐺 NeighbVtx 𝐶)))
67		eleq1 2825	. . . . . . 7 ⊢ (𝑛 = 𝐵 → (𝑛 ∈ (𝐺 NeighbVtx 𝐶) ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)))
68	66, 67	ralprg 4655	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (∀𝑛 ∈ {𝐴, 𝐵}𝑛 ∈ (𝐺 NeighbVtx 𝐶) ↔ (𝐴 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))))
69	68	3adant3 1132	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (∀𝑛 ∈ {𝐴, 𝐵}𝑛 ∈ (𝐺 NeighbVtx 𝐶) ↔ (𝐴 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))))
70	60, 65, 69	3anbi123d 1436	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → ((∀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)))))
71	70	3ad2ant1 1133	. . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ((∀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 1446	. . . 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))))
73	72	a1i 11	. . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 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))) ↔ ((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 28311	. . . . . . 7 ⊢ (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ 𝐴 ∈ (𝐺 NeighbVtx 𝐵))
75		nbgrsym 28311	. . . . . . 7 ⊢ (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))
76		nbgrsym 28311	. . . . . . 7 ⊢ (𝐴 ∈ (𝐺 NeighbVtx 𝐶) ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴))
77	74, 75, 76	3anbi123i 1155	. . . . . 6 ⊢ ((𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐶)) ↔ (𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
78	77	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ((𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐶)) ↔ (𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴))))
79	78	anbi1d 630	. . . 4 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 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))) ↔ ((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 1100	. . . . . . . 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)))
81	80	bicomi 223	. . . . . . 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)))
82	81	anbi1i 624	. . . . . 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 565	. . . . . 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)))
84	82, 83	bitri 274	. . . . 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)))
85	84	a1i 11	. . . 4 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 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))) ↔ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))))
86		tpid1g 4730	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ {𝐴, 𝐵, 𝐶})
87		tpid2g 4732	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑌 → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
88		tpid3g 4733	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑍 → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
89	86, 87, 88	3anim123i 1151	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
90		df-3an 1089	. . . . . . 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}))
91	89, 90	sylib 217	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → ((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
92		simplr 767	. . . . . . . . . 10 ⊢ (((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
93	92	anim1ci 616	. . . . . . . . 9 ⊢ ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph) → (𝐺 ∈ UPGraph ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}))
94	93	3adant3 1132	. . . . . . . 8 ⊢ ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (𝐺 ∈ UPGraph ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}))
95		simpll 765	. . . . . . . . 9 ⊢ (((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) → 𝐴 ∈ {𝐴, 𝐵, 𝐶})
96		simp1 1136	. . . . . . . . 9 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐵)
97	95, 96	anim12i 613	. . . . . . . 8 ⊢ ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐴 ≠ 𝐵))
98		cplgr3v.e	. . . . . . . . 9 ⊢ 𝐸 = (Edg‘𝐺)
99	2, 98	nbupgrel 28293	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐴 ≠ 𝐵)) → (𝐴 ∈ (𝐺 NeighbVtx 𝐵) ↔ {𝐴, 𝐵} ∈ 𝐸))
100	94, 97, 99	3imp3i2an 1345	. . . . . . 7 ⊢ ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (𝐴 ∈ (𝐺 NeighbVtx 𝐵) ↔ {𝐴, 𝐵} ∈ 𝐸))
101		simpr 485	. . . . . . . . . 10 ⊢ (((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
102	101	anim1ci 616	. . . . . . . . 9 ⊢ ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph) → (𝐺 ∈ UPGraph ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
103	102	3adant3 1132	. . . . . . . 8 ⊢ ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (𝐺 ∈ UPGraph ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
104		simp3 1138	. . . . . . . . 9 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐶)
105	92, 104	anim12i 613	. . . . . . . 8 ⊢ ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ≠ 𝐶))
106	2, 98	nbupgrel 28293	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ (𝐵 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ≠ 𝐶)) → (𝐵 ∈ (𝐺 NeighbVtx 𝐶) ↔ {𝐵, 𝐶} ∈ 𝐸))
107	103, 105, 106	3imp3i2an 1345	. . . . . . 7 ⊢ ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (𝐵 ∈ (𝐺 NeighbVtx 𝐶) ↔ {𝐵, 𝐶} ∈ 𝐸))
108	95	anim1ci 616	. . . . . . . . 9 ⊢ ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph) → (𝐺 ∈ UPGraph ∧ 𝐴 ∈ {𝐴, 𝐵, 𝐶}))
109	108	3adant3 1132	. . . . . . . 8 ⊢ ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (𝐺 ∈ UPGraph ∧ 𝐴 ∈ {𝐴, 𝐵, 𝐶}))
110		simp2 1137	. . . . . . . . . 10 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶)
111	110	necomd 2999	. . . . . . . . 9 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝐶 ≠ 𝐴)
112	101, 111	anim12i 613	. . . . . . . 8 ⊢ ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ≠ 𝐴))
113	2, 98	nbupgrel 28293	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐴 ∈ {𝐴, 𝐵, 𝐶}) ∧ (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ≠ 𝐴)) → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐶, 𝐴} ∈ 𝐸))
114	109, 112, 113	3imp3i2an 1345	. . . . . . 7 ⊢ ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐶, 𝐴} ∈ 𝐸))
115	100, 107, 114	3anbi123d 1436	. . . . . 6 ⊢ ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ((𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))
116	91, 115	syl3an1 1163	. . . . 5 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ((𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))
117	80, 116	bitrid 282	. . . 4 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ((𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))
118	79, 85, 117	3bitrd 304	. . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 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))) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))
119	71, 73, 118	3bitrd 304	. 2 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ((∀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)))
120	4, 56, 119	3bitrd 304	1 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → (𝐺 ∈ ComplGraph ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∖ cdif 3907 {csn 4586 {cpr 4588 {ctp 4590 ‘cfv 6496 (class class class)co 7357 Vtxcvtx 27947 Edgcedg 27998 UPGraphcupgr 28031 NeighbVtx cnbgr 28280 ComplGraphccplgr 28357

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-n0 12414 df-xnn0 12486 df-z 12500 df-uz 12764 df-fz 13425 df-hash 14231 df-edg 27999 df-upgr 28033 df-nbgr 28281 df-uvtx 28334 df-cplgr 28359

This theorem is referenced by: cusgr3vnbpr 28384

W3C validator