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Theorem vtxval0 26387
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 5389 . . 3 ¬ ∅ ∈ (V × V)
21iffalsei 4317 . 2 if(∅ ∈ (V × V), (1st ‘∅), (Base‘∅)) = (Base‘∅)
3 vtxval 26348 . 2 (Vtx‘∅) = if(∅ ∈ (V × V), (1st ‘∅), (Base‘∅))
4 base0 16308 . 2 ∅ = (Base‘∅)
52, 3, 43eqtr4i 2812 1 (Vtx‘∅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1601  wcel 2107  Vcvv 3398  c0 4141  ifcif 4307   × cxp 5353  cfv 6135  1st c1st 7443  Basecbs 16255  Vtxcvtx 26344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-slot 16259  df-base 16261  df-vtx 26346
This theorem is referenced by:  uhgr0  26421  usgr0  26590  0grsubgr  26625  cplgr0  26773  vtxdg0v  26821  0grrusgr  26927  0wlk0  27000  0conngr  27595  frgr0  27672
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