Theorem nb3grpr 27652

Description: The neighbors of a vertex in a simple graph with three elements are an unordered pair of the other vertices iff all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.) (Revised by AV, 28-Oct-2020.)

Hypotheses

Ref	Expression
nb3grpr.v	⊢ 𝑉 = (Vtx‘𝐺)
nb3grpr.e	⊢ 𝐸 = (Edg‘𝐺)
nb3grpr.g	⊢ (𝜑 → 𝐺 ∈ USGraph)
nb3grpr.t	⊢ (𝜑 → 𝑉 = {𝐴, 𝐵, 𝐶})
nb3grpr.s	⊢ (𝜑 → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍))
nb3grpr.n	⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))

Assertion

Ref	Expression
nb3grpr	⊢ (𝜑 → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ ∀𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧}))

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑦,𝐸   𝑥,𝐺,𝑦,𝑧   𝑥,𝑉,𝑦,𝑧   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑧)   𝐸(𝑥,𝑧)   𝑋(𝑥,𝑦,𝑧)   𝑌(𝑥,𝑦,𝑧)   𝑍(𝑥,𝑦,𝑧)

Proof of Theorem nb3grpr

Step	Hyp	Ref	Expression
1		id 22	. . . . . 6 ⊢ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸))
2		prcom 4665	. . . . . . . . . 10 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
3	2	eleq1i 2829	. . . . . . . . 9 ⊢ ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐵, 𝐴} ∈ 𝐸)
4		prcom 4665	. . . . . . . . . 10 ⊢ {𝐵, 𝐶} = {𝐶, 𝐵}
5	4	eleq1i 2829	. . . . . . . . 9 ⊢ ({𝐵, 𝐶} ∈ 𝐸 ↔ {𝐶, 𝐵} ∈ 𝐸)
6		prcom 4665	. . . . . . . . . 10 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
7	6	eleq1i 2829	. . . . . . . . 9 ⊢ ({𝐶, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐶} ∈ 𝐸)
8	3, 5, 7	3anbi123i 1153	. . . . . . . 8 ⊢ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ ({𝐵, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))
9		3anrot 1098	. . . . . . . 8 ⊢ (({𝐴, 𝐶} ∈ 𝐸 ∧ {𝐵, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸) ↔ ({𝐵, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))
10	8, 9	bitr4i 277	. . . . . . 7 ⊢ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ ({𝐴, 𝐶} ∈ 𝐸 ∧ {𝐵, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))
11	10	a1i 11	. . . . . 6 ⊢ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ ({𝐴, 𝐶} ∈ 𝐸 ∧ {𝐵, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
12	1, 11	biadanii 818	. . . . 5 ⊢ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ∧ ({𝐴, 𝐶} ∈ 𝐸 ∧ {𝐵, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
13		an6 1443	. . . . 5 ⊢ ((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ∧ ({𝐴, 𝐶} ∈ 𝐸 ∧ {𝐵, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)) ↔ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸) ∧ ({𝐵, 𝐶} ∈ 𝐸 ∧ {𝐵, 𝐴} ∈ 𝐸) ∧ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
14	12, 13	bitri 274	. . . 4 ⊢ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸) ∧ ({𝐵, 𝐶} ∈ 𝐸 ∧ {𝐵, 𝐴} ∈ 𝐸) ∧ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
15	14	a1i 11	. . 3 ⊢ (𝜑 → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸) ∧ ({𝐵, 𝐶} ∈ 𝐸 ∧ {𝐵, 𝐴} ∈ 𝐸) ∧ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))))
16		nb3grpr.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
17		nb3grpr.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
18		nb3grpr.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ USGraph)
19		nb3grpr.t	. . . . 5 ⊢ (𝜑 → 𝑉 = {𝐴, 𝐵, 𝐶})
20		nb3grpr.s	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍))
21	16, 17, 18, 19, 20	nb3grprlem1 27650	. . . 4 ⊢ (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)))
22		tprot 4682	. . . . . 6 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
23	19, 22	eqtrdi 2795	. . . . 5 ⊢ (𝜑 → 𝑉 = {𝐵, 𝐶, 𝐴})
24		3anrot 1098	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ↔ (𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋))
25	20, 24	sylib 217	. . . . 5 ⊢ (𝜑 → (𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋))
26	16, 17, 18, 23, 25	nb3grprlem1 27650	. . . 4 ⊢ (𝜑 → ((𝐺 NeighbVtx 𝐵) = {𝐶, 𝐴} ↔ ({𝐵, 𝐶} ∈ 𝐸 ∧ {𝐵, 𝐴} ∈ 𝐸)))
27		tprot 4682	. . . . . 6 ⊢ {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
28	19, 27	eqtr4di 2797	. . . . 5 ⊢ (𝜑 → 𝑉 = {𝐶, 𝐴, 𝐵})
29		3anrot 1098	. . . . . 6 ⊢ ((𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍))
30	20, 29	sylibr 233	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌))
31	16, 17, 18, 28, 30	nb3grprlem1 27650	. . . 4 ⊢ (𝜑 → ((𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵} ↔ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
32	21, 26, 31	3anbi123d 1434	. . 3 ⊢ (𝜑 → (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐶, 𝐴} ∧ (𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵}) ↔ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸) ∧ ({𝐵, 𝐶} ∈ 𝐸 ∧ {𝐵, 𝐴} ∈ 𝐸) ∧ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))))
33		nb3grpr.n	. . . . 5 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶))
34	16, 17, 18, 19, 20, 33	nb3grprlem2 27651	. . . 4 ⊢ (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐴) = {𝑦, 𝑧}))
35		necom 2996	. . . . . . . 8 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
36		necom 2996	. . . . . . . 8 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐶 ≠ 𝐴)
37		biid 260	. . . . . . . 8 ⊢ (𝐵 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)
38	35, 36, 37	3anbi123i 1153	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ↔ (𝐵 ≠ 𝐴 ∧ 𝐶 ≠ 𝐴 ∧ 𝐵 ≠ 𝐶))
39		3anrot 1098	. . . . . . 7 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴 ∧ 𝐶 ≠ 𝐴) ↔ (𝐵 ≠ 𝐴 ∧ 𝐶 ≠ 𝐴 ∧ 𝐵 ≠ 𝐶))
40	38, 39	bitr4i 277	. . . . . 6 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ↔ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴 ∧ 𝐶 ≠ 𝐴))
41	33, 40	sylib 217	. . . . 5 ⊢ (𝜑 → (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴 ∧ 𝐶 ≠ 𝐴))
42	16, 17, 18, 23, 25, 41	nb3grprlem2 27651	. . . 4 ⊢ (𝜑 → ((𝐺 NeighbVtx 𝐵) = {𝐶, 𝐴} ↔ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐵) = {𝑦, 𝑧}))
43		3anrot 1098	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ↔ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵))
44		necom 2996	. . . . . . . 8 ⊢ (𝐵 ≠ 𝐶 ↔ 𝐶 ≠ 𝐵)
45		biid 260	. . . . . . . 8 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐴 ≠ 𝐵)
46	36, 44, 45	3anbi123i 1153	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ↔ (𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵))
47	43, 46	bitri 274	. . . . . 6 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ↔ (𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵))
48	33, 47	sylib 217	. . . . 5 ⊢ (𝜑 → (𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵))
49	16, 17, 18, 28, 30, 48	nb3grprlem2 27651	. . . 4 ⊢ (𝜑 → ((𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵} ↔ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐶) = {𝑦, 𝑧}))
50	34, 42, 49	3anbi123d 1434	. . 3 ⊢ (𝜑 → (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐶, 𝐴} ∧ (𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵}) ↔ (∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐴) = {𝑦, 𝑧} ∧ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐵) = {𝑦, 𝑧} ∧ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐶) = {𝑦, 𝑧})))
51	15, 32, 50	3bitr2d 306	. 2 ⊢ (𝜑 → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ (∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐴) = {𝑦, 𝑧} ∧ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐵) = {𝑦, 𝑧} ∧ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐶) = {𝑦, 𝑧})))
52		oveq2 7263	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝐴))
53	52	eqeq1d 2740	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ (𝐺 NeighbVtx 𝐴) = {𝑦, 𝑧}))
54	53	2rexbidv 3228	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐴) = {𝑦, 𝑧}))
55		oveq2 7263	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝐵))
56	55	eqeq1d 2740	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ (𝐺 NeighbVtx 𝐵) = {𝑦, 𝑧}))
57	56	2rexbidv 3228	. . . 4 ⊢ (𝑥 = 𝐵 → (∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐵) = {𝑦, 𝑧}))
58		oveq2 7263	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝐶))
59	58	eqeq1d 2740	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ (𝐺 NeighbVtx 𝐶) = {𝑦, 𝑧}))
60	59	2rexbidv 3228	. . . 4 ⊢ (𝑥 = 𝐶 → (∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐶) = {𝑦, 𝑧}))
61	54, 57, 60	raltpg 4631	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ (∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐴) = {𝑦, 𝑧} ∧ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐵) = {𝑦, 𝑧} ∧ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐶) = {𝑦, 𝑧})))
62	20, 61	syl 17	. 2 ⊢ (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ (∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐴) = {𝑦, 𝑧} ∧ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐵) = {𝑦, 𝑧} ∧ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐶) = {𝑦, 𝑧})))
63		raleq 3333	. . . 4 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (∀𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧}))
64	63	bicomd 222	. . 3 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ ∀𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧}))
65	19, 64	syl 17	. 2 ⊢ (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ ∀𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧}))
66	51, 62, 65	3bitr2d 306	1 ⊢ (𝜑 → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ ∀𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064   ∖ cdif 3880  {csn 4558  {cpr 4560  {ctp 4562  ‘cfv 6418  (class class class)co 7255  Vtxcvtx 27269  Edgcedg 27320  USGraphcusgr 27422   NeighbVtx cnbgr 27602

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-umgr 27356  df-usgr 27424  df-nbgr 27603

This theorem is referenced by:  cusgr3vnbpr  27706