Theorem uvtxa0 40620

Description: There is no universal vertex if there is no vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.)

Hypothesis

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

Assertion

Ref	Expression
uvtxa0	⊢ (𝑉 = ∅ → (UnivVtx‘𝐺) = ∅)

Proof of Theorem uvtxa0

Dummy variables 𝑛 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uvtxael.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	uvtxaval 40613	. . . . 5 ⊢ (𝐺 ∈ V → (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)})
3	2	adantr 480	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑉 = ∅) → (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)})
4		rab0 3909	. . . . 5 ⊢ {𝑣 ∈ ∅ ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = ∅
5		rabeq 3166	. . . . . . 7 ⊢ (𝑉 = ∅ → {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = {𝑣 ∈ ∅ ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)})
6	5	eqeq1d 2612	. . . . . 6 ⊢ (𝑉 = ∅ → ({𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = ∅ ↔ {𝑣 ∈ ∅ ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = ∅))
7	6	adantl 481	. . . . 5 ⊢ ((𝐺 ∈ V ∧ 𝑉 = ∅) → ({𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = ∅ ↔ {𝑣 ∈ ∅ ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = ∅))
8	4, 7	mpbiri 247	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑉 = ∅) → {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = ∅)
9	3, 8	eqtrd 2644	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝑉 = ∅) → (UnivVtx‘𝐺) = ∅)
10	9	ex 449	. 2 ⊢ (𝐺 ∈ V → (𝑉 = ∅ → (UnivVtx‘𝐺) = ∅))
11		fvprc 6097	. . 3 ⊢ (¬ 𝐺 ∈ V → (UnivVtx‘𝐺) = ∅)
12	11	a1d 25	. 2 ⊢ (¬ 𝐺 ∈ V → (𝑉 = ∅ → (UnivVtx‘𝐺) = ∅))
13	10, 12	pm2.61i 175	1 ⊢ (𝑉 = ∅ → (UnivVtx‘𝐺) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  {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

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  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-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-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-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-uvtxa 40556

This theorem is referenced by: (None)