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Theorem cplgr3v 27705

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 2747	. . . 4 ⊢ {𝐴, 𝐵, 𝐶} = (Vtx‘𝐺)
3	2	iscplgrnb 27686	. . 3 ⊢ (𝐺 ∈ UPGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ {𝐴, 𝐵, 𝐶}∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
4	3	3ad2ant2 1132	. 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 4568	. . . . . 6 ⊢ (𝑣 = 𝐴 → {𝑣} = {𝐴})
6	5	difeq2d 4053	. . . . 5 ⊢ (𝑣 = 𝐴 → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐴}))
7		tprot 4682	. . . . . . . 8 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
8	7	difeq1i 4049	. . . . . . 7 ⊢ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) = ({𝐵, 𝐶, 𝐴} ∖ {𝐴})
9		necom 2996	. . . . . . . . 9 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
10		necom 2996	. . . . . . . . 9 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐶 ≠ 𝐴)
11		diftpsn3 4732	. . . . . . . . 9 ⊢ ((𝐵 ≠ 𝐴 ∧ 𝐶 ≠ 𝐴) → ({𝐵, 𝐶, 𝐴} ∖ {𝐴}) = {𝐵, 𝐶})
12	9, 10, 11	syl2anb 597	. . . . . . . 8 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ({𝐵, 𝐶, 𝐴} ∖ {𝐴}) = {𝐵, 𝐶})
13	12	3adant3 1130	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐵, 𝐶, 𝐴} ∖ {𝐴}) = {𝐵, 𝐶})
14	8, 13	syl5eq 2791	. . . . . 6 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) = {𝐵, 𝐶})
15	14	3ad2ant3 1133	. . . . 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 2801	. . . 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 7263	. . . . . 6 ⊢ (𝑣 = 𝐴 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝐴))
18	17	eleq2d 2824	. . . . 5 ⊢ (𝑣 = 𝐴 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝐴)))
19	18	adantl 481	. . . 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 3327	. . 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 4568	. . . . . 6 ⊢ (𝑣 = 𝐵 → {𝑣} = {𝐵})
22	21	difeq2d 4053	. . . . 5 ⊢ (𝑣 = 𝐵 → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐵}))
23		tprot 4682	. . . . . . . . 9 ⊢ {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
24	23	eqcomi 2747	. . . . . . . 8 ⊢ {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴, 𝐵}
25	24	difeq1i 4049	. . . . . . 7 ⊢ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) = ({𝐶, 𝐴, 𝐵} ∖ {𝐵})
26		necom 2996	. . . . . . . . . . . 12 ⊢ (𝐵 ≠ 𝐶 ↔ 𝐶 ≠ 𝐵)
27	26	biimpi 215	. . . . . . . . . . 11 ⊢ (𝐵 ≠ 𝐶 → 𝐶 ≠ 𝐵)
28	27	anim2i 616	. . . . . . . . . 10 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) → (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵))
29	28	ancomd 461	. . . . . . . . 9 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) → (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵))
30		diftpsn3 4732	. . . . . . . . 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 1129	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐶, 𝐴, 𝐵} ∖ {𝐵}) = {𝐶, 𝐴})
33	25, 32	syl5eq 2791	. . . . . 6 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) = {𝐶, 𝐴})
34	33	3ad2ant3 1133	. . . . 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 2801	. . . 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 7263	. . . . . 6 ⊢ (𝑣 = 𝐵 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝐵))
37	36	eleq2d 2824	. . . . 5 ⊢ (𝑣 = 𝐵 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝐵)))
38	37	adantl 481	. . . 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 3327	. . 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 4568	. . . . . 6 ⊢ (𝑣 = 𝐶 → {𝑣} = {𝐶})
41	40	difeq2d 4053	. . . . 5 ⊢ (𝑣 = 𝐶 → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐶}))
42		diftpsn3 4732	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
43	42	3adant1 1128	. . . . . 6 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
44	43	3ad2ant3 1133	. . . . 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 2801	. . . 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 7263	. . . . . 6 ⊢ (𝑣 = 𝐶 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝐶))
47	46	eleq2d 2824	. . . . 5 ⊢ (𝑣 = 𝐶 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝐶)))
48	47	adantl 481	. . . 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 3327	. . 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 1134	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐴 ∈ 𝑋)
51	50	3ad2ant1 1131	. . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → 𝐴 ∈ 𝑋)
52		simp2 1135	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐵 ∈ 𝑌)
53	52	3ad2ant1 1131	. . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → 𝐵 ∈ 𝑌)
54		simp3 1136	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 ∈ 𝑍)
55	54	3ad2ant1 1131	. . 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 4714	. 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 2826	. . . . . . 7 ⊢ (𝑛 = 𝐵 → (𝑛 ∈ (𝐺 NeighbVtx 𝐴) ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
58		eleq1 2826	. . . . . . 7 ⊢ (𝑛 = 𝐶 → (𝑛 ∈ (𝐺 NeighbVtx 𝐴) ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
59	57, 58	ralprg 4627	. . . . . 6 ⊢ ((𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (∀𝑛 ∈ {𝐵, 𝐶}𝑛 ∈ (𝐺 NeighbVtx 𝐴) ↔ (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴))))
60	59	3adant1 1128	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (∀𝑛 ∈ {𝐵, 𝐶}𝑛 ∈ (𝐺 NeighbVtx 𝐴) ↔ (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴))))
61		eleq1 2826	. . . . . . . 8 ⊢ (𝑛 = 𝐶 → (𝑛 ∈ (𝐺 NeighbVtx 𝐵) ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐵)))
62		eleq1 2826	. . . . . . . 8 ⊢ (𝑛 = 𝐴 → (𝑛 ∈ (𝐺 NeighbVtx 𝐵) ↔ 𝐴 ∈ (𝐺 NeighbVtx 𝐵)))
63	61, 62	ralprg 4627	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋) → (∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵) ↔ (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵))))
64	63	ancoms 458	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑍) → (∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵) ↔ (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵))))
65	64	3adant2 1129	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵) ↔ (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵))))
66		eleq1 2826	. . . . . . 7 ⊢ (𝑛 = 𝐴 → (𝑛 ∈ (𝐺 NeighbVtx 𝐶) ↔ 𝐴 ∈ (𝐺 NeighbVtx 𝐶)))
67		eleq1 2826	. . . . . . 7 ⊢ (𝑛 = 𝐵 → (𝑛 ∈ (𝐺 NeighbVtx 𝐶) ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)))
68	66, 67	ralprg 4627	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (∀𝑛 ∈ {𝐴, 𝐵}𝑛 ∈ (𝐺 NeighbVtx 𝐶) ↔ (𝐴 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))))
69	68	3adant3 1130	. . . . 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 1434	. . . 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 1131	. . 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 1444	. . . 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 27633	. . . . . . 7 ⊢ (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ 𝐴 ∈ (𝐺 NeighbVtx 𝐵))
75		nbgrsym 27633	. . . . . . 7 ⊢ (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))
76		nbgrsym 27633	. . . . . . 7 ⊢ (𝐴 ∈ (𝐺 NeighbVtx 𝐶) ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴))
77	74, 75, 76	3anbi123i 1153	. . . . . 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 629	. . . 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 1098	. . . . . . . 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 623	. . . . . 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 564	. . . . . 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 4702	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ {𝐴, 𝐵, 𝐶})
87		tpid2g 4704	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑌 → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
88		tpid3g 4705	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑍 → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
89	86, 87, 88	3anim123i 1149	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
90		df-3an 1087	. . . . . . 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 765	. . . . . . . . . 10 ⊢ (((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
93	92	anim1ci 615	. . . . . . . . 9 ⊢ ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph) → (𝐺 ∈ UPGraph ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}))
94	93	3adant3 1130	. . . . . . . 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 763	. . . . . . . . 9 ⊢ (((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) → 𝐴 ∈ {𝐴, 𝐵, 𝐶})
96		simp1 1134	. . . . . . . . 9 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐵)
97	95, 96	anim12i 612	. . . . . . . 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 27615	. . . . . . . 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 1343	. . . . . . 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 484	. . . . . . . . . 10 ⊢ (((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
102	101	anim1ci 615	. . . . . . . . 9 ⊢ ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph) → (𝐺 ∈ UPGraph ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
103	102	3adant3 1130	. . . . . . . 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 1136	. . . . . . . . 9 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐶)
105	92, 104	anim12i 612	. . . . . . . 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 27615	. . . . . . . 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 1343	. . . . . . 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 615	. . . . . . . . 9 ⊢ ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph) → (𝐺 ∈ UPGraph ∧ 𝐴 ∈ {𝐴, 𝐵, 𝐶}))
109	108	3adant3 1130	. . . . . . . 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 1135	. . . . . . . . . 10 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶)
111	110	necomd 2998	. . . . . . . . 9 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝐶 ≠ 𝐴)
112	101, 111	anim12i 612	. . . . . . . 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 27615	. . . . . . . 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 1343	. . . . . . 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 1434	. . . . . 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 1161	. . . . 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	syl5bb 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 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∖ cdif 3880 {csn 4558 {cpr 4560 {ctp 4562 ‘cfv 6418 (class class class)co 7255 Vtxcvtx 27269 Edgcedg 27320 UPGraphcupgr 27353 NeighbVtx cnbgr 27602 ComplGraphccplgr 27679

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-n0 12164 df-xnn0 12236 df-z 12250 df-uz 12512 df-fz 13169 df-hash 13973 df-edg 27321 df-upgr 27355 df-nbgr 27603 df-uvtx 27656 df-cplgr 27681

This theorem is referenced by: cusgr3vnbpr 27706

W3C validator