Theorem uvtxnbgrb 40628

Description: A vertex is universal iff all the other vertices are its neighbors. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 23-Mar-2021.)

Hypothesis

Ref	Expression
uvtxnbgr.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
uvtxnbgrb	⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})))

Proof of Theorem uvtxnbgrb

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uvtxnbgr.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	uvtxnbgr 40627	. 2 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
3		simpl 472	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) → 𝑁 ∈ 𝑉)
4		raleleq 3133	. . . . . 6 ⊢ ((𝑉 ∖ {𝑁}) = (𝐺 NeighbVtx 𝑁) → ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))
5	4	eqcoms 2618	. . . . 5 ⊢ ((𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}) → ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))
6	5	adantl 481	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) → ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))
7	1	1vgrex 25679	. . . . . 6 ⊢ (𝑁 ∈ 𝑉 → 𝐺 ∈ V)
8	7	adantr 480	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) → 𝐺 ∈ V)
9	1	uvtxael 40614	. . . . 5 ⊢ (𝐺 ∈ V → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))))
10	8, 9	syl 17	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))))
11	3, 6, 10	mpbir2and 959	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) → 𝑁 ∈ (UnivVtx‘𝐺))
12	11	ex 449	. 2 ⊢ (𝑁 ∈ 𝑉 → ((𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}) → 𝑁 ∈ (UnivVtx‘𝐺)))
13	2, 12	impbid2 215	1 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∖ cdif 3537  {csn 4125  ‘cfv 5804  (class class class)co 6549  Vtxcvtx 25673   NeighbVtx cnbgr 40550  UnivVtxcuvtxa 40551

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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  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-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-nbgr 40554  df-uvtxa 40556

This theorem is referenced by:  nbusgrvtxm1uvtx  40632  uvtxupgrres  40635  nbcplgr  40656