Theorem uvtxnbgrb 29418

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.) (Proof shortened by AV, 14-Feb-2022.)

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 29417	. 2 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
3		simpl 482	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) → 𝑁 ∈ 𝑉)
4		raleleq 3342	. . . . . 6 ⊢ ((𝑉 ∖ {𝑁}) = (𝐺 NeighbVtx 𝑁) → ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))
5	4	eqcoms 2745	. . . . 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	uvtxel 29405	. . . 4 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)))
8	3, 6, 7	sylanbrc 583	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) → 𝑁 ∈ (UnivVtx‘𝐺))
9	8	ex 412	. 2 ⊢ (𝑁 ∈ 𝑉 → ((𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}) → 𝑁 ∈ (UnivVtx‘𝐺)))
10	2, 9	impbid2 226	1 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061   ∖ cdif 3948  {csn 4626  ‘cfv 6561  (class class class)co 7431  Vtxcvtx 29013   NeighbVtx cnbgr 29349  UnivVtxcuvtx 29402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-nbgr 29350  df-uvtx 29403

This theorem is referenced by:  nbusgrvtxm1uvtx  29422  uvtxupgrres  29425  nbcplgr  29451