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Theorem vtxval0 39185
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 4881 . . 3  |-  -.  (/)  e.  ( _V  X.  _V )
21iffalsei 3903 . 2  |-  if (
(/)  e.  ( _V  X.  _V ) ,  ( 1st `  (/) ) ,  ( Base `  (/) ) )  =  ( Base `  (/) )
3 0ex 4549 . . 3  |-  (/)  e.  _V
4 vtxval 39151 . . 3  |-  ( (/)  e.  _V  ->  (Vtx `  (/) )  =  if ( (/)  e.  ( _V  X.  _V ) ,  ( 1st `  (/) ) ,  ( Base `  (/) ) ) )
53, 4ax-mp 5 . 2  |-  (Vtx `  (/) )  =  if (
(/)  e.  ( _V  X.  _V ) ,  ( 1st `  (/) ) ,  ( Base `  (/) ) )
6 base0 15211 . 2  |-  (/)  =  (
Base `  (/) )
72, 5, 63eqtr4i 2494 1  |-  (Vtx `  (/) )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1455    e. wcel 1898   _Vcvv 3057   (/)c0 3743   ifcif 3893    X. cxp 4851   ` cfv 5601   1stc1st 6818   Basecbs 15170  Vtxcvtx 39147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-slot 15174  df-base 15175  df-vtx 39149
This theorem is referenced by:  uhgr0  39213  usgr0  39368  0grsubgr  39400  cplgr0  39543  vtxdg0v  39583  0grrusgr  39645  1wlkv0  39830  0conngr  39933
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