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Theorem vtxval0 26130
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 5300 . . 3 ¬ ∅ ∈ (V × V)
21iffalsei 4240 . 2 if(∅ ∈ (V × V), (1st ‘∅), (Base‘∅)) = (Base‘∅)
3 vtxval 26077 . 2 (Vtx‘∅) = if(∅ ∈ (V × V), (1st ‘∅), (Base‘∅))
4 base0 16114 . 2 ∅ = (Base‘∅)
52, 3, 43eqtr4i 2792 1 (Vtx‘∅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  wcel 2139  Vcvv 3340  c0 4058  ifcif 4230   × cxp 5264  cfv 6049  1st c1st 7331  Basecbs 16059  Vtxcvtx 26073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-slot 16063  df-base 16065  df-vtx 26075
This theorem is referenced by:  uhgr0  26167  usgr0  26334  0grsubgr  26369  cplgr0  26531  vtxdg0v  26579  0grrusgr  26685  0wlk0  26759  0conngr  27344  frgr0  27418
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