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Theorem usgrav 24459
Description: The classes of vertices and edges of an undirected simple graph without loops are sets. (Contributed by Alexander van der Vekens, 19-Aug-2017.)
Assertion
Ref Expression
usgrav  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )

Proof of Theorem usgrav
StepHypRef Expression
1 relusgra 24456 . . 3  |-  Rel USGrph
21brrelexi 4954 . 2  |-  ( V USGrph  E  ->  V  e.  _V )
31brrelex2i 4955 . 2  |-  ( V USGrph  E  ->  E  e.  _V )
42, 3jca 530 1  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1826   _Vcvv 3034   class class class wbr 4367   USGrph cusg 24451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-xp 4919  df-rel 4920  df-usgra 24454
This theorem is referenced by:  usgraf  24467  usgraf0  24469  usisuslgra  24486  usgrares  24490  usgra0v  24492  usgra1v  24511  usgraedgleord  24515  fiusgraedgfi  24528  usgrafisindb0  24529  usgrafisindb1  24530  usgrares1  24531  usgrafisinds  24534  usgrafisbase  24535  nbusgra  24549  nbgrasym  24560  nbgraf1olem1  24562  cusgra2v  24583  nbcusgra  24584  cusgra3v  24585  cusgrares  24593  cusgrasize2inds  24598  cusgrafi  24603  sizeusglecusg  24607  uvtxnbgra  24614  cusgrauvtxb  24617  uvtxnb  24618  usgra2adedgspth  24734  usgra2adedgwlk  24735  usgra2adedgwlkon  24736  constr3trl  24780  constr3pth  24781  constr3cycl  24782  cusconngra  24797  clwwlkgt0  24892  clwwlkn0  24895  clwwlkn2  24896  clwlkisclwwlk  24910  hashclwwlkn  24957  clwwlkndivn  24958  clwlkfclwwlk  24965  usg2wlkonot  25004  usg2wotspth  25005  2pthwlkonot  25006  usg2spthonot0  25010  usg2spthonot1  25011  vdusgraval  25028  vdgrnn0pnf  25030  hashnbgravdg  25034  usgravd0nedg  25039  cusgraisrusgra  25059  0eusgraiff0rgra  25060  rusgraprop3  25064  rusgrasn  25066  rusgranumwwlkl1  25067  rusgranumwlkl1  25068  rusgranumwlkb0  25074  rusgra0edg  25076  frgra1v  25119  frgra2v  25120  frgra3v  25123  2pthfrgra  25132  3cyclfrgrarn  25134  frconngra  25142  vdfrgra0  25143  vdgn0frgrav2  25145  vdgn1frgrav2  25147  vdgfrgragt2  25148  frgrawopreglem1  25165  frg2wot1  25178  frghash2spot  25184  usg2spot2nb  25186  usgreg2spot  25188  frgraregorufrg  25193  extwwlkfablem1  25195  extwwlkfablem2  25199  numclwwlkovfel2  25204  numclwwlkovf2ex  25207  numclwwlkovgelim  25210  numclwwlk1  25219  numclwwlkqhash  25221  numclwwlk3lem  25229  numclwwlk3  25230  numclwwlk4  25231  numclwwlk5  25233  numclwwlk6  25234  frgrareg  25238  usgra2pth  32672  usgedgffibi  32752  usgrafisbaseALT  32758  usgrafisbaseALT2  32759  usgfis  32764  usgfisALT  32768
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