Theorem fusgrusgr 26214

Description: A finite simple graph is a simple graph. (Contributed by AV, 16-Jan-2020.) (Revised by AV, 21-Oct-2020.)

Assertion

Ref	Expression
fusgrusgr	|- ( G e. FinUSGraph -> G e. USGraph )

Proof of Theorem fusgrusgr

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- (Vtx ` G ) = (Vtx ` G )
2	1	isfusgr 26210	. 2 |- ( G e. FinUSGraph <-> ( G e. USGraph /\ (Vtx ` G ) e. Fin ) )
3	2	simplbi 476	1 |- ( G e. FinUSGraph -> G e. USGraph )

Syntax hints:   -> wi 4   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:  fusgredgfi  26217  fusgrfisstep  26221  fusgrfupgrfs  26223  nbfiusgrfi  26277  vtxdgfusgrf  26393  usgruvtxvdb  26425  vdiscusgrb  26426  vdiscusgr  26427  fusgrn0eqdrusgr  26466  wlksnfi  26802  fusgrhashclwwlkn  26956  clwlksfclwwlk  26962  clwlksfoclwwlk  26963  clwlksf1clwwlk  26969  fusgr2wsp2nb  27198  fusgreghash2wspv  27199  numclwwlk4  27244