Theorem nbuhgr2vtx1edgb 26248

Description: If a hypergraph 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.)

Hypotheses

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

Assertion

Ref	Expression
nbuhgr2vtx1edgb	⊢ ((𝐺 ∈ UHGraph ∧ (#‘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 nbuhgr2vtx1edgb

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		fvex 6201	. . . . 5 ⊢ (Vtx‘𝐺) ∈ V
3	1, 2	eqeltri 2697	. . . 4 ⊢ 𝑉 ∈ V
4		hash2prb 13254	. . . 4 ⊢ (𝑉 ∈ V → ((#‘𝑉) = 2 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})))
5	3, 4	ax-mp 5	. . 3 ⊢ ((#‘𝑉) = 2 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}))
6		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))
7	6	ancomd 467	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉))
8	7	ad2antrr 762	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉))
9		id 22	. . . . . . . . . . . . 13 ⊢ (𝑎 ≠ 𝑏 → 𝑎 ≠ 𝑏)
10	9	necomd 2849	. . . . . . . . . . . 12 ⊢ (𝑎 ≠ 𝑏 → 𝑏 ≠ 𝑎)
11	10	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → 𝑏 ≠ 𝑎)
12	11	ad2antlr 763	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑏 ≠ 𝑎)
13		prcom 4267	. . . . . . . . . . . . . 14 ⊢ {𝑎, 𝑏} = {𝑏, 𝑎}
14	13	eleq1i 2692	. . . . . . . . . . . . 13 ⊢ ({𝑎, 𝑏} ∈ 𝐸 ↔ {𝑏, 𝑎} ∈ 𝐸)
15	14	biimpi 206	. . . . . . . . . . . 12 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → {𝑏, 𝑎} ∈ 𝐸)
16		sseq2 3627	. . . . . . . . . . . . 13 ⊢ (𝑒 = {𝑏, 𝑎} → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ {𝑏, 𝑎}))
17	16	adantl 482	. . . . . . . . . . . 12 ⊢ (({𝑎, 𝑏} ∈ 𝐸 ∧ 𝑒 = {𝑏, 𝑎}) → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ {𝑏, 𝑎}))
18	13	eqimssi 3659	. . . . . . . . . . . . 13 ⊢ {𝑎, 𝑏} ⊆ {𝑏, 𝑎}
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → {𝑎, 𝑏} ⊆ {𝑏, 𝑎})
20	15, 17, 19	rspcedvd 3317	. . . . . . . . . . 11 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒)
21	20	adantl 482	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒)
22		nbgr2vtx1edg.e	. . . . . . . . . . . 12 ⊢ 𝐸 = (Edg‘𝐺)
23	1, 22	nbgrel 26238	. . . . . . . . . . 11 ⊢ (𝐺 ∈ UHGraph → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ ((𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ≠ 𝑎 ∧ ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒)))
24	23	ad3antrrr 766	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ ((𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ≠ 𝑎 ∧ ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒)))
25	8, 12, 21, 24	mpbir3and 1245	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑏 ∈ (𝐺 NeighbVtx 𝑎))
26	6	ad2antrr 762	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))
27		simplrl 800	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑎 ≠ 𝑏)
28		id 22	. . . . . . . . . . . 12 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → {𝑎, 𝑏} ∈ 𝐸)
29		sseq2 3627	. . . . . . . . . . . . 13 ⊢ (𝑒 = {𝑎, 𝑏} → ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑏, 𝑎} ⊆ {𝑎, 𝑏}))
30	29	adantl 482	. . . . . . . . . . . 12 ⊢ (({𝑎, 𝑏} ∈ 𝐸 ∧ 𝑒 = {𝑎, 𝑏}) → ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑏, 𝑎} ⊆ {𝑎, 𝑏}))
31		prcom 4267	. . . . . . . . . . . . . 14 ⊢ {𝑏, 𝑎} = {𝑎, 𝑏}
32	31	eqimssi 3659	. . . . . . . . . . . . 13 ⊢ {𝑏, 𝑎} ⊆ {𝑎, 𝑏}
33	32	a1i 11	. . . . . . . . . . . 12 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → {𝑏, 𝑎} ⊆ {𝑎, 𝑏})
34	28, 30, 33	rspcedvd 3317	. . . . . . . . . . 11 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒)
35	34	adantl 482	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒)
36	1, 22	nbgrel 26238	. . . . . . . . . . 11 ⊢ (𝐺 ∈ UHGraph → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏 ∧ ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒)))
37	36	ad3antrrr 766	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏 ∧ ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒)))
38	26, 27, 35, 37	mpbir3and 1245	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑎 ∈ (𝐺 NeighbVtx 𝑏))
39	25, 38	jca 554	. . . . . . . 8 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
40	39	ex 450	. . . . . . 7 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) → ({𝑎, 𝑏} ∈ 𝐸 → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
41	1, 22	nbuhgr2vtx1edgblem 26247	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏} ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)) → {𝑎, 𝑏} ∈ 𝐸)
42	41	3exp 1264	. . . . . . . . . . 11 ⊢ (𝐺 ∈ UHGraph → (𝑉 = {𝑎, 𝑏} → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
43	42	adantr 481	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑉 = {𝑎, 𝑏} → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
44	43	adantld 483	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
45	44	imp 445	. . . . . . . 8 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸))
46	45	adantld 483	. . . . . . 7 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) → ((𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)) → {𝑎, 𝑏} ∈ 𝐸))
47	40, 46	impbid 202	. . . . . 6 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) → ({𝑎, 𝑏} ∈ 𝐸 ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
48		eleq1 2689	. . . . . . . . 9 ⊢ (𝑉 = {𝑎, 𝑏} → (𝑉 ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
49	48	adantl 482	. . . . . . . 8 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝑉 ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
50		id 22	. . . . . . . . . 10 ⊢ (𝑉 = {𝑎, 𝑏} → 𝑉 = {𝑎, 𝑏})
51		difeq1 3721	. . . . . . . . . . 11 ⊢ (𝑉 = {𝑎, 𝑏} → (𝑉 ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑣}))
52	51	raleqdv 3144	. . . . . . . . . 10 ⊢ (𝑉 = {𝑎, 𝑏} → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
53	50, 52	raleqbidv 3152	. . . . . . . . 9 ⊢ (𝑉 = {𝑎, 𝑏} → (∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑣 ∈ {𝑎, 𝑏}∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
54		vex 3203	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
55		vex 3203	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
56		sneq 4187	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑎 → {𝑣} = {𝑎})
57	56	difeq2d 3728	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑎 → ({𝑎, 𝑏} ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑎}))
58		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑎 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑎))
59	58	eleq2d 2687	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑎 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
60	57, 59	raleqbidv 3152	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑎 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
61		sneq 4187	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑏 → {𝑣} = {𝑏})
62	61	difeq2d 3728	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑏 → ({𝑎, 𝑏} ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑏}))
63		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑏 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑏))
64	63	eleq2d 2687	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑏 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
65	62, 64	raleqbidv 3152	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
66	54, 55, 60, 65	ralpr 4238	. . . . . . . . . 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)))
67		difprsn1 4330	. . . . . . . . . . . . 13 ⊢ (𝑎 ≠ 𝑏 → ({𝑎, 𝑏} ∖ {𝑎}) = {𝑏})
68	67	raleqdv 3144	. . . . . . . . . . . 12 ⊢ (𝑎 ≠ 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ ∀𝑛 ∈ {𝑏}𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
69		eleq1 2689	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑏 → (𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)))
70	55, 69	ralsn 4222	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ {𝑏}𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎))
71	68, 70	syl6bb 276	. . . . . . . . . . 11 ⊢ (𝑎 ≠ 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)))
72		difprsn2 4331	. . . . . . . . . . . . 13 ⊢ (𝑎 ≠ 𝑏 → ({𝑎, 𝑏} ∖ {𝑏}) = {𝑎})
73	72	raleqdv 3144	. . . . . . . . . . . 12 ⊢ (𝑎 ≠ 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ ∀𝑛 ∈ {𝑎}𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
74		eleq1 2689	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑎 → (𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
75	54, 74	ralsn 4222	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ {𝑎}𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))
76	73, 75	syl6bb 276	. . . . . . . . . . 11 ⊢ (𝑎 ≠ 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
77	71, 76	anbi12d 747	. . . . . . . . . 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))))
78	66, 77	syl5bb 272	. . . . . . . . 9 ⊢ (𝑎 ≠ 𝑏 → (∀𝑣 ∈ {𝑎, 𝑏}∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
79	53, 78	sylan9bbr 737	. . . . . . . 8 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
80	49, 79	bibi12d 335	. . . . . . 7 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → ((𝑉 ∈ 𝐸 ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)) ↔ ({𝑎, 𝑏} ∈ 𝐸 ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))))
81	80	adantl 482	. . . . . 6 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) → ((𝑉 ∈ 𝐸 ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)) ↔ ({𝑎, 𝑏} ∈ 𝐸 ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))))
82	47, 81	mpbird 247	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) → (𝑉 ∈ 𝐸 ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
83	82	ex 450	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝑉 ∈ 𝐸 ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
84	83	rexlimdvva 3038	. . 3 ⊢ (𝐺 ∈ UHGraph → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝑉 ∈ 𝐸 ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
85	5, 84	syl5bi 232	. 2 ⊢ (𝐺 ∈ UHGraph → ((#‘𝑉) = 2 → (𝑉 ∈ 𝐸 ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
86	85	imp 445	1 ⊢ ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 2) → (𝑉 ∈ 𝐸 ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  {csn 4177  {cpr 4179  ‘cfv 5888  (class class class)co 6650  2c2 11070  #chash 13117  Vtxcvtx 25874  Edgcedg 25939   UHGraph cuhgr 25951   NeighbVtx cnbgr 26224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-edg 25940  df-uhgr 25953  df-nbgr 26228

This theorem is referenced by:  uvtx2vtx1edgb  26300