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Theorem vtxval0 25714
 Description: Degenerated case 1 for vertices: The set of vertices of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.)
Assertion
Ref Expression
vtxval0 (Vtx‘∅) = ∅

Proof of Theorem vtxval0
StepHypRef Expression
1 0nelxp 5067 . . 3 ¬ ∅ ∈ (V × V)
21iffalsei 4046 . 2 if(∅ ∈ (V × V), (1st ‘∅), (Base‘∅)) = (Base‘∅)
3 0ex 4718 . . 3 ∅ ∈ V
4 vtxval 25677 . . 3 (∅ ∈ V → (Vtx‘∅) = if(∅ ∈ (V × V), (1st ‘∅), (Base‘∅)))
53, 4ax-mp 5 . 2 (Vtx‘∅) = if(∅ ∈ (V × V), (1st ‘∅), (Base‘∅))
6 base0 15740 . 2 ∅ = (Base‘∅)
72, 5, 63eqtr4i 2642 1 (Vtx‘∅) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  ifcif 4036   × cxp 5036  ‘cfv 5804  1st c1st 7057  Basecbs 15695  Vtxcvtx 25673 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-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  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-slot 15699  df-base 15700  df-vtx 25675 This theorem is referenced by:  uhgr0  25739  usgr0  40469  0grsubgr  40502  cplgr0  40647  vtxdg0v  40688  0grrusgr  40779  01wlk0  40861  0conngr  41359  frgr0  41436
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