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Theorem 0vgrargra 30503
 Description: A graph with no vertices is k-regular for every k. (Contributed by Alexander van der Vekens, 10-Jul-2018.)
Assertion
Ref Expression
0vgrargra RegGrph
Distinct variable groups:   ,   ,

Proof of Theorem 0vgrargra
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . 3
2 ral0 3779 . . . 4 VDeg
32a1i 11 . . 3 VDeg
4 0ex 4417 . . . 4
5 isrgra 30496 . . . 4 RegGrph VDeg
64, 5mp3an1 1301 . . 3 RegGrph VDeg
71, 3, 6mpbir2and 913 . 2 RegGrph
87ralrimiva 2794 1 RegGrph
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1369   wcel 1756  wral 2710  cvv 2967  c0 3632  cop 3878   class class class wbr 4287  cfv 5413  (class class class)co 6086  cn0 10571   VDeg cvdg 23514   RegGrph crgra 30492 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-iota 5376  df-fv 5421  df-ov 6089  df-oprab 6090  df-rgra 30494 This theorem is referenced by: (None)
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