Theorem fusgrvtxfi 26211

Description: A finite simple graph has a finite set of vertices. (Contributed by AV, 16-Dec-2020.)

Hypothesis

Ref	Expression
isfusgr.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
fusgrvtxfi	|- ( G e. FinUSGraph -> V e. Fin )

Proof of Theorem fusgrvtxfi

Step	Hyp	Ref	Expression
1		isfusgr.v	. . 3 |- V = (Vtx ` G )
2	1	isfusgr 26210	. 2 |- ( G e. FinUSGraph <-> ( G e. USGraph /\ V e. Fin ) )
3	2	simprbi 480	1 |- ( G e. FinUSGraph -> V e. Fin )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888   Fincfn 7955  Vtxcvtx 25874   USGraph cusgr 26044   FinUSGraph cfusgr 26208

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  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-iota 5851  df-fv 5896  df-fusgr 26209

This theorem is referenced by:  fusgrfupgrfs  26223  nbusgrvtxm1  26281  uvtxanm1nbgr  26305  cusgrm1rusgr  26478  wlksnfi  26802  fusgrhashclwwlkn  26956  clwwlksndivn  26957  fusgreghash2wsp  27202  numclwwlk3lem  27241  numclwwlk4  27244