Theorem uvtxa0 26294

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 26287	. . . . 5 ⊢ (𝐺 ∈ V → (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)})
3	2	adantr 481	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑉 = ∅) → (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)})
4		rab0 3955	. . . . 5 ⊢ {𝑣 ∈ ∅ ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = ∅
5		rabeq 3192	. . . . . . 7 ⊢ (𝑉 = ∅ → {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = {𝑣 ∈ ∅ ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)})
6	5	eqeq1d 2624	. . . . . 6 ⊢ (𝑉 = ∅ → ({𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = ∅ ↔ {𝑣 ∈ ∅ ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = ∅))
7	6	adantl 482	. . . . 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 248	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑉 = ∅) → {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = ∅)
9	3, 8	eqtrd 2656	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝑉 = ∅) → (UnivVtx‘𝐺) = ∅)
10	9	ex 450	. 2 ⊢ (𝐺 ∈ V → (𝑉 = ∅ → (UnivVtx‘𝐺) = ∅))
11		fvprc 6185	. . 3 ⊢ (¬ 𝐺 ∈ V → (UnivVtx‘𝐺) = ∅)
12	11	a1d 25	. 2 ⊢ (¬ 𝐺 ∈ V → (𝑉 = ∅ → (UnivVtx‘𝐺) = ∅))
13	10, 12	pm2.61i 176	1 ⊢ (𝑉 = ∅ → (UnivVtx‘𝐺) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200   ∖ cdif 3571  ∅c0 3915  {csn 4177  ‘cfv 5888  (class class class)co 6650  Vtxcvtx 25874   NeighbVtx cnbgr 26224  UnivVtxcuvtxa 26225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-uvtxa 26230

This theorem is referenced by: (None)