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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  |-  ( E  e.  V  ->  A. k  e.  NN0  <. (/) ,  E >. RegGrph  k )
Distinct variable groups:    k, E    k, V

Proof of Theorem 0vgrargra
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . 3  |-  ( ( E  e.  V  /\  k  e.  NN0 )  -> 
k  e.  NN0 )
2 ral0 3779 . . . 4  |-  A. p  e.  (/)  ( ( (/) VDeg  E ) `  p )  =  k
32a1i 11 . . 3  |-  ( ( E  e.  V  /\  k  e.  NN0 )  ->  A. p  e.  (/)  ( (
(/) VDeg  E ) `  p
)  =  k )
4 0ex 4417 . . . 4  |-  (/)  e.  _V
5 isrgra 30496 . . . 4  |-  ( (
(/)  e.  _V  /\  E  e.  V  /\  k  e.  NN0 )  ->  ( <.
(/) ,  E >. RegGrph  k  <-> 
( k  e.  NN0  /\ 
A. p  e.  (/)  ( ( (/) VDeg  E ) `
 p )  =  k ) ) )
64, 5mp3an1 1301 . . 3  |-  ( ( E  e.  V  /\  k  e.  NN0 )  -> 
( <. (/) ,  E >. RegGrph  k  <-> 
( k  e.  NN0  /\ 
A. p  e.  (/)  ( ( (/) VDeg  E ) `
 p )  =  k ) ) )
71, 3, 6mpbir2and 913 . 2  |-  ( ( E  e.  V  /\  k  e.  NN0 )  ->  <.
(/) ,  E >. RegGrph  k )
87ralrimiva 2794 1  |-  ( E  e.  V  ->  A. k  e.  NN0  <. (/) ,  E >. RegGrph  k )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2710   _Vcvv 2967   (/)c0 3632   <.cop 3878   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   NN0cn0 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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