Theorem nbuhgr2vtx1edgb 29374

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.) (Proof shortened by AV, 13-Feb-2022.)

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	1	fvexi 6846	. . . 4 ⊢ 𝑉 ∈ V
3		hash2prb 14393	. . . 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		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))
6	5	ancomd 461	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉))
7	6	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉))
8		id 22	. . . . . . . . . . . . 13 ⊢ (𝑎 ≠ 𝑏 → 𝑎 ≠ 𝑏)
9	8	necomd 2985	. . . . . . . . . . . 12 ⊢ (𝑎 ≠ 𝑏 → 𝑏 ≠ 𝑎)
10	9	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → 𝑏 ≠ 𝑎)
11	10	ad2antlr 727	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑏 ≠ 𝑎)
12		prcom 4687	. . . . . . . . . . . . . 14 ⊢ {𝑎, 𝑏} = {𝑏, 𝑎}
13	12	eleq1i 2825	. . . . . . . . . . . . 13 ⊢ ({𝑎, 𝑏} ∈ 𝐸 ↔ {𝑏, 𝑎} ∈ 𝐸)
14	13	biimpi 216	. . . . . . . . . . . 12 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → {𝑏, 𝑎} ∈ 𝐸)
15		sseq2 3958	. . . . . . . . . . . . 13 ⊢ (𝑒 = {𝑏, 𝑎} → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ {𝑏, 𝑎}))
16	15	adantl 481	. . . . . . . . . . . 12 ⊢ (({𝑎, 𝑏} ∈ 𝐸 ∧ 𝑒 = {𝑏, 𝑎}) → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ {𝑏, 𝑎}))
17	12	eqimssi 3992	. . . . . . . . . . . . 13 ⊢ {𝑎, 𝑏} ⊆ {𝑏, 𝑎}
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → {𝑎, 𝑏} ⊆ {𝑏, 𝑎})
19	14, 16, 18	rspcedvd 3576	. . . . . . . . . . 11 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒)
20	19	adantl 481	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒)
21		nbgr2vtx1edg.e	. . . . . . . . . . 11 ⊢ 𝐸 = (Edg‘𝐺)
22	1, 21	nbgrel 29362	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ ((𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ≠ 𝑎 ∧ ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒))
23	7, 11, 20, 22	syl3anbrc 1344	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑏 ∈ (𝐺 NeighbVtx 𝑎))
24	5	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))
25		simplrl 776	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑎 ≠ 𝑏)
26		id 22	. . . . . . . . . . . 12 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → {𝑎, 𝑏} ∈ 𝐸)
27		sseq2 3958	. . . . . . . . . . . . 13 ⊢ (𝑒 = {𝑎, 𝑏} → ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑏, 𝑎} ⊆ {𝑎, 𝑏}))
28	27	adantl 481	. . . . . . . . . . . 12 ⊢ (({𝑎, 𝑏} ∈ 𝐸 ∧ 𝑒 = {𝑎, 𝑏}) → ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑏, 𝑎} ⊆ {𝑎, 𝑏}))
29		prcom 4687	. . . . . . . . . . . . . 14 ⊢ {𝑏, 𝑎} = {𝑎, 𝑏}
30	29	eqimssi 3992	. . . . . . . . . . . . 13 ⊢ {𝑏, 𝑎} ⊆ {𝑎, 𝑏}
31	30	a1i 11	. . . . . . . . . . . 12 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → {𝑏, 𝑎} ⊆ {𝑎, 𝑏})
32	26, 28, 31	rspcedvd 3576	. . . . . . . . . . 11 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒)
33	32	adantl 481	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒)
34	1, 21	nbgrel 29362	. . . . . . . . . 10 ⊢ (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏 ∧ ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒))
35	24, 25, 33, 34	syl3anbrc 1344	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑎 ∈ (𝐺 NeighbVtx 𝑏))
36	23, 35	jca 511	. . . . . . . 8 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
37	36	ex 412	. . . . . . 7 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) → ({𝑎, 𝑏} ∈ 𝐸 → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
38	1, 21	nbuhgr2vtx1edgblem 29373	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏} ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)) → {𝑎, 𝑏} ∈ 𝐸)
39	38	3exp 1119	. . . . . . . . . . 11 ⊢ (𝐺 ∈ UHGraph → (𝑉 = {𝑎, 𝑏} → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
40	39	adantr 480	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑉 = {𝑎, 𝑏} → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
41	40	adantld 490	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
42	41	imp 406	. . . . . . . 8 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸))
43	42	adantld 490	. . . . . . 7 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) → ((𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)) → {𝑎, 𝑏} ∈ 𝐸))
44	37, 43	impbid 212	. . . . . 6 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) → ({𝑎, 𝑏} ∈ 𝐸 ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
45		eleq1 2822	. . . . . . . . 9 ⊢ (𝑉 = {𝑎, 𝑏} → (𝑉 ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
46	45	adantl 481	. . . . . . . 8 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝑉 ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
47		id 22	. . . . . . . . . 10 ⊢ (𝑉 = {𝑎, 𝑏} → 𝑉 = {𝑎, 𝑏})
48		difeq1 4069	. . . . . . . . . . 11 ⊢ (𝑉 = {𝑎, 𝑏} → (𝑉 ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑣}))
49	48	raleqdv 3294	. . . . . . . . . 10 ⊢ (𝑉 = {𝑎, 𝑏} → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
50	47, 49	raleqbidv 3314	. . . . . . . . 9 ⊢ (𝑉 = {𝑎, 𝑏} → (∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑣 ∈ {𝑎, 𝑏}∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
51		vex 3442	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
52		vex 3442	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
53		sneq 4588	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑎 → {𝑣} = {𝑎})
54	53	difeq2d 4076	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑎 → ({𝑎, 𝑏} ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑎}))
55		oveq2 7364	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑎 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑎))
56	55	eleq2d 2820	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑎 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
57	54, 56	raleqbidv 3314	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑎 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
58		sneq 4588	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑏 → {𝑣} = {𝑏})
59	58	difeq2d 4076	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑏 → ({𝑎, 𝑏} ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑏}))
60		oveq2 7364	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑏 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑏))
61	60	eleq2d 2820	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑏 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
62	59, 61	raleqbidv 3314	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
63	51, 52, 57, 62	ralpr 4655	. . . . . . . . . 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)))
64		difprsn1 4754	. . . . . . . . . . . . 13 ⊢ (𝑎 ≠ 𝑏 → ({𝑎, 𝑏} ∖ {𝑎}) = {𝑏})
65	64	raleqdv 3294	. . . . . . . . . . . 12 ⊢ (𝑎 ≠ 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ ∀𝑛 ∈ {𝑏}𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
66		eleq1 2822	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑏 → (𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)))
67	52, 66	ralsn 4636	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ {𝑏}𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎))
68	65, 67	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝑎 ≠ 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)))
69		difprsn2 4755	. . . . . . . . . . . . 13 ⊢ (𝑎 ≠ 𝑏 → ({𝑎, 𝑏} ∖ {𝑏}) = {𝑎})
70	69	raleqdv 3294	. . . . . . . . . . . 12 ⊢ (𝑎 ≠ 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ ∀𝑛 ∈ {𝑎}𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
71		eleq1 2822	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑎 → (𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
72	51, 71	ralsn 4636	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ {𝑎}𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))
73	70, 72	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝑎 ≠ 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
74	68, 73	anbi12d 632	. . . . . . . . . 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))))
75	63, 74	bitrid 283	. . . . . . . . 9 ⊢ (𝑎 ≠ 𝑏 → (∀𝑣 ∈ {𝑎, 𝑏}∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
76	50, 75	sylan9bbr 510	. . . . . . . 8 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
77	46, 76	bibi12d 345	. . . . . . 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)))))
78	77	adantl 481	. . . . . 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)))))
79	44, 78	mpbird 257	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) → (𝑉 ∈ 𝐸 ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
80	79	ex 412	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝑉 ∈ 𝐸 ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
81	80	rexlimdvva 3191	. . 3 ⊢ (𝐺 ∈ UHGraph → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝑉 ∈ 𝐸 ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
82	4, 81	biimtrid 242	. 2 ⊢ (𝐺 ∈ UHGraph → ((♯‘𝑉) = 2 → (𝑉 ∈ 𝐸 ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
83	82	imp 406	1 ⊢ ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 2) → (𝑉 ∈ 𝐸 ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049  ∃wrex 3058  Vcvv 3438   ∖ cdif 3896   ⊆ wss 3899  {csn 4578  {cpr 4580  ‘cfv 6490  (class class class)co 7356  2c2 12198  ♯chash 14251  Vtxcvtx 29018  Edgcedg 29069  UHGraphcuhgr 29078   NeighbVtx cnbgr 29354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-dju 9811  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-n0 12400  df-z 12487  df-uz 12750  df-fz 13422  df-hash 14252  df-edg 29070  df-uhgr 29080  df-nbgr 29355

This theorem is referenced by:  uvtx2vtx1edgb  29421