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Theorem 0vgrargra 25641
 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 462 . . 3
2 ral0 3899 . . . 4 VDeg
32a1i 11 . . 3 VDeg
4 0ex 4549 . . . 4
5 isrgra 25630 . . . 4 RegGrph VDeg
64, 5mp3an1 1347 . . 3 RegGrph VDeg
71, 3, 6mpbir2and 930 . 2 RegGrph
87ralrimiva 2837 1 RegGrph
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370   wceq 1437   wcel 1867  wral 2773  cvv 3078  c0 3758  cop 3999   class class class wbr 4417  cfv 5593  (class class class)co 6297  cn0 10865   VDeg cvdg 25597   RegGrph crgra 25626 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pr 4653 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-iota 5557  df-fv 5601  df-ov 6300  df-oprab 6301  df-rgra 25628 This theorem is referenced by: (None)
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