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Theorem 0vgrargra 30718
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 3895 . . . 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 4533 . . . 4  |-  (/)  e.  _V
5 isrgra 30711 . . . 4  |-  ( (
(/)  e.  _V  /\  E  e.  V  /\  k  e.  NN0 )  ->  ( <.
(/) ,  E >. RegGrph  k  <-> 
( k  e.  NN0  /\ 
A. p  e.  (/)  ( ( (/) VDeg  E ) `
 p )  =  k ) ) )
64, 5mp3an1 1302 . . 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 2830 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 1370    e. wcel 1758   A.wral 2799   _Vcvv 3078   (/)c0 3748   <.cop 3994   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   NN0cn0 10693   VDeg cvdg 23735   RegGrph crgra 30707
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 4524  ax-nul 4532  ax-pr 4642
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-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-iota 5492  df-fv 5537  df-ov 6206  df-oprab 6207  df-rgra 30709
This theorem is referenced by: (None)
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