Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rusgrargra Structured version   Unicode version

Theorem rusgrargra 25228
 Description: A k-regular undirected simple graph is a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
Assertion
Ref Expression
rusgrargra RegUSGrph RegGrph

Proof of Theorem rusgrargra
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rusgra 25223 . . . 4 RegUSGrph USGrph RegGrph
21breqi 4400 . . 3 RegUSGrph USGrph RegGrph
3 oprabv 6282 . . 3 USGrph RegGrph
42, 3sylbi 195 . 2 RegUSGrph
5 isrusgra 25225 . . 3 RegUSGrph USGrph VDeg
6 isrgra 25224 . . . . . 6 RegGrph VDeg
76biimprcd 225 . . . . 5 VDeg RegGrph
873adant1 1015 . . . 4 USGrph VDeg RegGrph
98com12 29 . . 3 USGrph VDeg RegGrph
105, 9sylbid 215 . 2 RegUSGrph RegGrph
114, 10mpcom 34 1 RegUSGrph RegGrph
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 367   w3a 974   wceq 1405   wcel 1842  wral 2753  cvv 3058  cop 3977   class class class wbr 4394  cfv 5525  (class class class)co 6234  coprab 6235  cn0 10756   USGrph cusg 24628   VDeg cvdg 25191   RegGrph crgra 25220   RegUSGrph crusgra 25221 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-xp 4948  df-rel 4949  df-iota 5489  df-fv 5533  df-ov 6237  df-oprab 6238  df-rgra 25222  df-rusgra 25223 This theorem is referenced by:  rusgra0edg  25253
 Copyright terms: Public domain W3C validator