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Theorem rusgraprop 31423
Description: The properties of a k-regular undirected simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
Assertion
Ref Expression
rusgraprop
Distinct variable groups:   ,   ,   ,

Proof of Theorem rusgraprop
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rusgra 31419 . . . 4
21breqi 4415 . . 3
3 oprabv 31034 . . 3
42, 3sylbi 195 . 2
5 isrusgra 31421 . . 3
65biimpd 207 . 2
74, 6mpcom 36 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  /\w3a 965  =wceq 1370  e.wcel 1758  A.wral 2800   cvv 3081  <.cop 3999   class class class wbr 4409  `cfv 5537  (class class class)co 6222  {coprab 6223   cn0 10717   cusg 23733   cvdg 24032   crgra 31416   crusgra 31417
This theorem is referenced by:  rusisusgra  31425  cusgraiffrusgra  31430  rusgraprop2  31431  rusgranumwlks  31451  rusgranumwlk  31452  frrusgraord  31541  numclwwlk3  31579  numclwwlk5lem  31581  numclwwlk5  31582  numclwwlk7  31584  frgrareggt1  31586  frgrareg  31587  frgraregord013  31588
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 1955  ax-ext 2432  ax-sep 4530  ax-nul 4538  ax-pr 4648
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3083  df-sbc 3298  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3752  df-if 3906  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4209  df-br 4410  df-opab 4468  df-xp 4963  df-rel 4964  df-iota 5500  df-fv 5545  df-ov 6225  df-oprab 6226  df-rgra 31418  df-rusgra 31419
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