Theorem nbuhgr2vtx1edgb 27719

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 6788	. . . 4 ⊢ 𝑉 ∈ V
3		hash2prb 14186	. . . 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 485	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))
6	5	ancomd 462	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉))
7	6	ad2antrr 723	. . . . . . . . . 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 2999	. . . . . . . . . . . 12 ⊢ (𝑎 ≠ 𝑏 → 𝑏 ≠ 𝑎)
10	9	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → 𝑏 ≠ 𝑎)
11	10	ad2antlr 724	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑏 ≠ 𝑎)
12		prcom 4668	. . . . . . . . . . . . . 14 ⊢ {𝑎, 𝑏} = {𝑏, 𝑎}
13	12	eleq1i 2829	. . . . . . . . . . . . 13 ⊢ ({𝑎, 𝑏} ∈ 𝐸 ↔ {𝑏, 𝑎} ∈ 𝐸)
14	13	biimpi 215	. . . . . . . . . . . 12 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → {𝑏, 𝑎} ∈ 𝐸)
15		sseq2 3947	. . . . . . . . . . . . 13 ⊢ (𝑒 = {𝑏, 𝑎} → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ {𝑏, 𝑎}))
16	15	adantl 482	. . . . . . . . . . . 12 ⊢ (({𝑎, 𝑏} ∈ 𝐸 ∧ 𝑒 = {𝑏, 𝑎}) → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ {𝑏, 𝑎}))
17	12	eqimssi 3979	. . . . . . . . . . . . 13 ⊢ {𝑎, 𝑏} ⊆ {𝑏, 𝑎}
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → {𝑎, 𝑏} ⊆ {𝑏, 𝑎})
19	14, 16, 18	rspcedvd 3563	. . . . . . . . . . 11 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒)
20	19	adantl 482	. . . . . . . . . 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 27707	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ ((𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ≠ 𝑎 ∧ ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒))
23	7, 11, 20, 22	syl3anbrc 1342	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑏 ∈ (𝐺 NeighbVtx 𝑎))
24	5	ad2antrr 723	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))
25		simplrl 774	. . . . . . . . . 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 3947	. . . . . . . . . . . . 13 ⊢ (𝑒 = {𝑎, 𝑏} → ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑏, 𝑎} ⊆ {𝑎, 𝑏}))
28	27	adantl 482	. . . . . . . . . . . 12 ⊢ (({𝑎, 𝑏} ∈ 𝐸 ∧ 𝑒 = {𝑎, 𝑏}) → ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑏, 𝑎} ⊆ {𝑎, 𝑏}))
29		prcom 4668	. . . . . . . . . . . . . 14 ⊢ {𝑏, 𝑎} = {𝑎, 𝑏}
30	29	eqimssi 3979	. . . . . . . . . . . . 13 ⊢ {𝑏, 𝑎} ⊆ {𝑎, 𝑏}
31	30	a1i 11	. . . . . . . . . . . 12 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → {𝑏, 𝑎} ⊆ {𝑎, 𝑏})
32	26, 28, 31	rspcedvd 3563	. . . . . . . . . . 11 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒)
33	32	adantl 482	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒)
34	1, 21	nbgrel 27707	. . . . . . . . . 10 ⊢ (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏 ∧ ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒))
35	24, 25, 33, 34	syl3anbrc 1342	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑎 ∈ (𝐺 NeighbVtx 𝑏))
36	23, 35	jca 512	. . . . . . . 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 413	. . . . . . 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 27718	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏} ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)) → {𝑎, 𝑏} ∈ 𝐸)
39	38	3exp 1118	. . . . . . . . . . 11 ⊢ (𝐺 ∈ UHGraph → (𝑉 = {𝑎, 𝑏} → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
40	39	adantr 481	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑉 = {𝑎, 𝑏} → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
41	40	adantld 491	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
42	41	imp 407	. . . . . . . 8 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸))
43	42	adantld 491	. . . . . . 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 211	. . . . . 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 2826	. . . . . . . . 9 ⊢ (𝑉 = {𝑎, 𝑏} → (𝑉 ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
46	45	adantl 482	. . . . . . . 8 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝑉 ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
47		id 22	. . . . . . . . . 10 ⊢ (𝑉 = {𝑎, 𝑏} → 𝑉 = {𝑎, 𝑏})
48		difeq1 4050	. . . . . . . . . . 11 ⊢ (𝑉 = {𝑎, 𝑏} → (𝑉 ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑣}))
49	48	raleqdv 3348	. . . . . . . . . 10 ⊢ (𝑉 = {𝑎, 𝑏} → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
50	47, 49	raleqbidv 3336	. . . . . . . . 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 3436	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
52		vex 3436	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
53		sneq 4571	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑎 → {𝑣} = {𝑎})
54	53	difeq2d 4057	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑎 → ({𝑎, 𝑏} ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑎}))
55		oveq2 7283	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑎 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑎))
56	55	eleq2d 2824	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑎 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
57	54, 56	raleqbidv 3336	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑎 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
58		sneq 4571	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑏 → {𝑣} = {𝑏})
59	58	difeq2d 4057	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑏 → ({𝑎, 𝑏} ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑏}))
60		oveq2 7283	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑏 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑏))
61	60	eleq2d 2824	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑏 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
62	59, 61	raleqbidv 3336	. . . . . . . . . . 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 4636	. . . . . . . . . 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 4733	. . . . . . . . . . . . 13 ⊢ (𝑎 ≠ 𝑏 → ({𝑎, 𝑏} ∖ {𝑎}) = {𝑏})
65	64	raleqdv 3348	. . . . . . . . . . . 12 ⊢ (𝑎 ≠ 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ ∀𝑛 ∈ {𝑏}𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
66		eleq1 2826	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑏 → (𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)))
67	52, 66	ralsn 4617	. . . . . . . . . . . 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 4734	. . . . . . . . . . . . 13 ⊢ (𝑎 ≠ 𝑏 → ({𝑎, 𝑏} ∖ {𝑏}) = {𝑎})
70	69	raleqdv 3348	. . . . . . . . . . . 12 ⊢ (𝑎 ≠ 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ ∀𝑛 ∈ {𝑎}𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
71		eleq1 2826	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑎 → (𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
72	51, 71	ralsn 4617	. . . . . . . . . . . 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 631	. . . . . . . . . 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 282	. . . . . . . . 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 511	. . . . . . . 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 346	. . . . . . 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 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)))))
79	44, 78	mpbird 256	. . . . 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 413	. . . 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 3223	. . 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	syl5bi 241	. 2 ⊢ (𝐺 ∈ UHGraph → ((♯‘𝑉) = 2 → (𝑉 ∈ 𝐸 ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
83	82	imp 407	1 ⊢ ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 2) → (𝑉 ∈ 𝐸 ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∖ cdif 3884   ⊆ wss 3887  {csn 4561  {cpr 4563  ‘cfv 6433  (class class class)co 7275  2c2 12028  ♯chash 14044  Vtxcvtx 27366  Edgcedg 27417  UHGraphcuhgr 27426   NeighbVtx cnbgr 27699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-oadd 8301  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-hash 14045  df-edg 27418  df-uhgr 27428  df-nbgr 27700

This theorem is referenced by:  uvtx2vtx1edgb  27766