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Theorem rusisusgra 30689
Description: Any k-regular undirected simple graph is an undirected simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
Assertion
Ref Expression
rusisusgra  |-  ( <. V ,  E >. RegUSGrph  K  ->  V USGrph  E )

Proof of Theorem rusisusgra
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 rusgraprop 30687 . 2  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. p  e.  V  (
( V VDeg  E ) `  p )  =  K ) )
21simp1d 1000 1  |-  ( <. V ,  E >. RegUSGrph  K  ->  V USGrph  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   A.wral 2795   <.cop 3984   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   NN0cn0 10683   USGrph cusg 23409   VDeg cvdg 23708   RegUSGrph crusgra 30681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-xp 4947  df-rel 4948  df-iota 5482  df-fv 5527  df-ov 6196  df-oprab 6197  df-rgra 30682  df-rusgra 30683
This theorem is referenced by:  rusgranumwlkl1lem1  30699  rusgranumwlkl1  30700  rusgranumwlkb1  30713  rusgra0edg  30714  rusgranumwlks  30715  rusgranumwlk  30716  rusgranumwwlkg  30718  numclwwlkovf2num  30819  numclwwlk1  30832  numclwwlkqhash  30834  numclwwlk3  30843  numclwwlk5  30846  numclwwlk6  30847  frgrareg  30851
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