Theorem nbgr2vtx1edg 29435

Description: If a graph has two vertices, and there is an edge between the vertices, then each vertex is the neighbor of the other vertex. (Contributed by AV, 2-Nov-2020.) (Revised by AV, 25-Mar-2021.) (Proof shortened by AV, 13-Feb-2022.)

Hypotheses

Ref	Expression
nbgr2vtx1edg.v	⊢ 𝑉 = (Vtx‘𝐺)
nbgr2vtx1edg.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
nbgr2vtx1edg	⊢ (((♯‘𝑉) = 2 ∧ 𝑉 ∈ 𝐸) → ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))

Distinct variable groups:   𝑛,𝐸   𝑛,𝐺,𝑣   𝑛,𝑉,𝑣

Allowed substitution hint:   𝐸(𝑣)

Proof of Theorem nbgr2vtx1edg

Dummy variables 𝑎 𝑏 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nbgr2vtx1edg.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	fvexi 6856	. . . 4 ⊢ 𝑉 ∈ V
3		hash2prb 14407	. . . 4 ⊢ (𝑉 ∈ V → ((♯‘𝑉) = 2 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})))
4	2, 3	ax-mp 5	. . 3 ⊢ ((♯‘𝑉) = 2 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}))
5		simpll 767	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))
6	5	ancomd 461	. . . . . . . . 9 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉))
7		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → 𝑎 ≠ 𝑏)
8	7	necomd 2988	. . . . . . . . . 10 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → 𝑏 ≠ 𝑎)
9	8	ad2antlr 728	. . . . . . . . 9 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑏 ≠ 𝑎)
10		id 22	. . . . . . . . . . 11 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → {𝑎, 𝑏} ∈ 𝐸)
11		sseq2 3962	. . . . . . . . . . . 12 ⊢ (𝑒 = {𝑎, 𝑏} → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ {𝑎, 𝑏}))
12	11	adantl 481	. . . . . . . . . . 11 ⊢ (({𝑎, 𝑏} ∈ 𝐸 ∧ 𝑒 = {𝑎, 𝑏}) → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ {𝑎, 𝑏}))
13		ssidd 3959	. . . . . . . . . . 11 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → {𝑎, 𝑏} ⊆ {𝑎, 𝑏})
14	10, 12, 13	rspcedvd 3580	. . . . . . . . . 10 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒)
15	14	adantl 481	. . . . . . . . 9 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒)
16		nbgr2vtx1edg.e	. . . . . . . . . 10 ⊢ 𝐸 = (Edg‘𝐺)
17	1, 16	nbgrel 29425	. . . . . . . . 9 ⊢ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ ((𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ≠ 𝑎 ∧ ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒))
18	6, 9, 15, 17	syl3anbrc 1345	. . . . . . . 8 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑏 ∈ (𝐺 NeighbVtx 𝑎))
19	7	ad2antlr 728	. . . . . . . . 9 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑎 ≠ 𝑏)
20		sseq2 3962	. . . . . . . . . . . 12 ⊢ (𝑒 = {𝑎, 𝑏} → ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑏, 𝑎} ⊆ {𝑎, 𝑏}))
21	20	adantl 481	. . . . . . . . . . 11 ⊢ (({𝑎, 𝑏} ∈ 𝐸 ∧ 𝑒 = {𝑎, 𝑏}) → ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑏, 𝑎} ⊆ {𝑎, 𝑏}))
22		prcom 4691	. . . . . . . . . . . . 13 ⊢ {𝑏, 𝑎} = {𝑎, 𝑏}
23	22	eqimssi 3996	. . . . . . . . . . . 12 ⊢ {𝑏, 𝑎} ⊆ {𝑎, 𝑏}
24	23	a1i 11	. . . . . . . . . . 11 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → {𝑏, 𝑎} ⊆ {𝑎, 𝑏})
25	10, 21, 24	rspcedvd 3580	. . . . . . . . . 10 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒)
26	25	adantl 481	. . . . . . . . 9 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒)
27	1, 16	nbgrel 29425	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏 ∧ ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒))
28	5, 19, 26, 27	syl3anbrc 1345	. . . . . . . 8 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑎 ∈ (𝐺 NeighbVtx 𝑏))
29		difprsn1 4758	. . . . . . . . . . . . 13 ⊢ (𝑎 ≠ 𝑏 → ({𝑎, 𝑏} ∖ {𝑎}) = {𝑏})
30	29	raleqdv 3298	. . . . . . . . . . . 12 ⊢ (𝑎 ≠ 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ ∀𝑛 ∈ {𝑏}𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
31		vex 3446	. . . . . . . . . . . . 13 ⊢ 𝑏 ∈ V
32		eleq1 2825	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑏 → (𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)))
33	31, 32	ralsn 4640	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ {𝑏}𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎))
34	30, 33	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝑎 ≠ 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)))
35		difprsn2 4759	. . . . . . . . . . . . 13 ⊢ (𝑎 ≠ 𝑏 → ({𝑎, 𝑏} ∖ {𝑏}) = {𝑎})
36	35	raleqdv 3298	. . . . . . . . . . . 12 ⊢ (𝑎 ≠ 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ ∀𝑛 ∈ {𝑎}𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
37		vex 3446	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
38		eleq1 2825	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑎 → (𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
39	37, 38	ralsn 4640	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ {𝑎}𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))
40	36, 39	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝑎 ≠ 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
41	34, 40	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑎 ≠ 𝑏 → ((∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)) ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
42	41	adantr 480	. . . . . . . . 9 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → ((∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)) ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
43	42	ad2antlr 728	. . . . . . . 8 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → ((∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)) ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
44	18, 28, 43	mpbir2and 714	. . . . . . 7 ⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
45	44	ex 412	. . . . . 6 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) → ({𝑎, 𝑏} ∈ 𝐸 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏))))
46		eleq1 2825	. . . . . . . . 9 ⊢ (𝑉 = {𝑎, 𝑏} → (𝑉 ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
47		id 22	. . . . . . . . . . 11 ⊢ (𝑉 = {𝑎, 𝑏} → 𝑉 = {𝑎, 𝑏})
48		difeq1 4073	. . . . . . . . . . . 12 ⊢ (𝑉 = {𝑎, 𝑏} → (𝑉 ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑣}))
49	48	raleqdv 3298	. . . . . . . . . . 11 ⊢ (𝑉 = {𝑎, 𝑏} → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
50	47, 49	raleqbidv 3318	. . . . . . . . . 10 ⊢ (𝑉 = {𝑎, 𝑏} → (∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑣 ∈ {𝑎, 𝑏}∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
51		sneq 4592	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑎 → {𝑣} = {𝑎})
52	51	difeq2d 4080	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑎 → ({𝑎, 𝑏} ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑎}))
53		oveq2 7376	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑎 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑎))
54	53	eleq2d 2823	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑎 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
55	52, 54	raleqbidv 3318	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑎 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
56		sneq 4592	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑏 → {𝑣} = {𝑏})
57	56	difeq2d 4080	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑏 → ({𝑎, 𝑏} ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑏}))
58		oveq2 7376	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑏 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑏))
59	58	eleq2d 2823	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑏 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
60	57, 59	raleqbidv 3318	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
61	37, 31, 55, 60	ralpr 4659	. . . . . . . . . 10 ⊢ (∀𝑣 ∈ {𝑎, 𝑏}∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
62	50, 61	bitrdi 287	. . . . . . . . 9 ⊢ (𝑉 = {𝑎, 𝑏} → (∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏))))
63	46, 62	imbi12d 344	. . . . . . . 8 ⊢ (𝑉 = {𝑎, 𝑏} → ((𝑉 ∈ 𝐸 → ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)) ↔ ({𝑎, 𝑏} ∈ 𝐸 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)))))
64	63	adantl 481	. . . . . . 7 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → ((𝑉 ∈ 𝐸 → ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)) ↔ ({𝑎, 𝑏} ∈ 𝐸 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)))))
65	64	adantl 481	. . . . . 6 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) → ((𝑉 ∈ 𝐸 → ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)) ↔ ({𝑎, 𝑏} ∈ 𝐸 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)))))
66	45, 65	mpbird 257	. . . . 5 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) → (𝑉 ∈ 𝐸 → ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
67	66	ex 412	. . . 4 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝑉 ∈ 𝐸 → ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
68	67	rexlimivv 3180	. . 3 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝑉 ∈ 𝐸 → ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
69	4, 68	sylbi 217	. 2 ⊢ ((♯‘𝑉) = 2 → (𝑉 ∈ 𝐸 → ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
70	69	imp 406	1 ⊢ (((♯‘𝑉) = 2 ∧ 𝑉 ∈ 𝐸) → ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3442   ∖ cdif 3900   ⊆ wss 3903  {csn 4582  {cpr 4584  ‘cfv 6500  (class class class)co 7368  2c2 12212  ♯chash 14265  Vtxcvtx 29081  Edgcedg 29132   NeighbVtx cnbgr 29417

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-dju 9825  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-n0 12414  df-z 12501  df-uz 12764  df-fz 13436  df-hash 14266  df-nbgr 29418

This theorem is referenced by:  uvtx2vtx1edg  29483