Theorem nb3grpr 40610

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 4211	. . . . . . . . . 10 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
3	2	eleq1i 2679	. . . . . . . . 9 ⊢ ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐵, 𝐴} ∈ 𝐸)
4		prcom 4211	. . . . . . . . . 10 ⊢ {𝐵, 𝐶} = {𝐶, 𝐵}
5	4	eleq1i 2679	. . . . . . . . 9 ⊢ ({𝐵, 𝐶} ∈ 𝐸 ↔ {𝐶, 𝐵} ∈ 𝐸)
6		prcom 4211	. . . . . . . . . 10 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
7	6	eleq1i 2679	. . . . . . . . 9 ⊢ ({𝐶, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐶} ∈ 𝐸)
8	3, 5, 7	3anbi123i 1244	. . . . . . . 8 ⊢ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ ({𝐵, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))
9		3anrot 1036	. . . . . . . 8 ⊢ (({𝐴, 𝐶} ∈ 𝐸 ∧ {𝐵, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸) ↔ ({𝐵, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))
10	8, 9	bitr4i 266	. . . . . . 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	biadan2 672	. . . . 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 1400	. . . . 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 263	. . . 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 40608	. . . 4 ⊢ (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)))
22		tprot 4228	. . . . . 6 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
23	19, 22	syl6eq 2660	. . . . 5 ⊢ (𝜑 → 𝑉 = {𝐵, 𝐶, 𝐴})
24		3anrot 1036	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ↔ (𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋))
25	20, 24	sylib 207	. . . . 5 ⊢ (𝜑 → (𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋))
26	16, 17, 18, 23, 25	nb3grprlem1 40608	. . . 4 ⊢ (𝜑 → ((𝐺 NeighbVtx 𝐵) = {𝐶, 𝐴} ↔ ({𝐵, 𝐶} ∈ 𝐸 ∧ {𝐵, 𝐴} ∈ 𝐸)))
27		tprot 4228	. . . . . 6 ⊢ {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
28	19, 27	syl6eqr 2662	. . . . 5 ⊢ (𝜑 → 𝑉 = {𝐶, 𝐴, 𝐵})
29		3anrot 1036	. . . . . 6 ⊢ ((𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍))
30	20, 29	sylibr 223	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌))
31	16, 17, 18, 28, 30	nb3grprlem1 40608	. . . 4 ⊢ (𝜑 → ((𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵} ↔ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
32	21, 26, 31	3anbi123d 1391	. . 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 40609	. . . 4 ⊢ (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐴) = {𝑦, 𝑧}))
35		necom 2835	. . . . . . . 8 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
36		necom 2835	. . . . . . . 8 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐶 ≠ 𝐴)
37		biid 250	. . . . . . . 8 ⊢ (𝐵 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)
38	35, 36, 37	3anbi123i 1244	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ↔ (𝐵 ≠ 𝐴 ∧ 𝐶 ≠ 𝐴 ∧ 𝐵 ≠ 𝐶))
39		3anrot 1036	. . . . . . 7 ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴 ∧ 𝐶 ≠ 𝐴) ↔ (𝐵 ≠ 𝐴 ∧ 𝐶 ≠ 𝐴 ∧ 𝐵 ≠ 𝐶))
40	38, 39	bitr4i 266	. . . . . 6 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ↔ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴 ∧ 𝐶 ≠ 𝐴))
41	33, 40	sylib 207	. . . . 5 ⊢ (𝜑 → (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐴 ∧ 𝐶 ≠ 𝐴))
42	16, 17, 18, 23, 25, 41	nb3grprlem2 40609	. . . 4 ⊢ (𝜑 → ((𝐺 NeighbVtx 𝐵) = {𝐶, 𝐴} ↔ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐵) = {𝑦, 𝑧}))
43		3anrot 1036	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ↔ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵))
44		necom 2835	. . . . . . . 8 ⊢ (𝐵 ≠ 𝐶 ↔ 𝐶 ≠ 𝐵)
45		biid 250	. . . . . . . 8 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐴 ≠ 𝐵)
46	36, 44, 45	3anbi123i 1244	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ↔ (𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵))
47	43, 46	bitri 263	. . . . . 6 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ↔ (𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵))
48	33, 47	sylib 207	. . . . 5 ⊢ (𝜑 → (𝐶 ≠ 𝐴 ∧ 𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵))
49	16, 17, 18, 28, 30, 48	nb3grprlem2 40609	. . . 4 ⊢ (𝜑 → ((𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵} ↔ ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝐶) = {𝑦, 𝑧}))
50	34, 42, 49	3anbi123d 1391	. . 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 295	. 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 6557	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝐴))
53	52	eqeq1d 2612	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ (𝐺 NeighbVtx 𝐴) = {𝑦, 𝑧}))
54	53	2rexbidv 3039	. . . 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 6557	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝐵))
56	55	eqeq1d 2612	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ (𝐺 NeighbVtx 𝐵) = {𝑦, 𝑧}))
57	56	2rexbidv 3039	. . . 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 6557	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝐶))
59	58	eqeq1d 2612	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧} ↔ (𝐺 NeighbVtx 𝐶) = {𝑦, 𝑧}))
60	59	2rexbidv 3039	. . . 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 4183	. . 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 3115	. . . 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 212	. . 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 295	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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537  {csn 4125  {cpr 4127  {ctp 4129  ‘cfv 5804  (class class class)co 6549  Vtxcvtx 25673  Edgcedga 25792   USGraph cusgr 40379   NeighbVtx cnbgr 40550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-upgr 25749  df-umgr 25750  df-edga 25793  df-usgr 40381  df-nbgr 40554

This theorem is referenced by:  cusgr3vnbpr  40658