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Theorem 0vgrargra 25341
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 459 . . 3  |-  ( ( E  e.  V  /\  k  e.  NN0 )  -> 
k  e.  NN0 )
2 ral0 3877 . . . 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 4525 . . . 4  |-  (/)  e.  _V
5 isrgra 25330 . . . 4  |-  ( (
(/)  e.  _V  /\  E  e.  V  /\  k  e.  NN0 )  ->  ( <.
(/) ,  E >. RegGrph  k  <-> 
( k  e.  NN0  /\ 
A. p  e.  (/)  ( ( (/) VDeg  E ) `
 p )  =  k ) ) )
64, 5mp3an1 1313 . . 3  |-  ( ( E  e.  V  /\  k  e.  NN0 )  -> 
( <. (/) ,  E >. RegGrph  k  <-> 
( k  e.  NN0  /\ 
A. p  e.  (/)  ( ( (/) VDeg  E ) `
 p )  =  k ) ) )
71, 3, 6mpbir2and 923 . 2  |-  ( ( E  e.  V  /\  k  e.  NN0 )  ->  <.
(/) ,  E >. RegGrph  k )
87ralrimiva 2817 1  |-  ( E  e.  V  ->  A. k  e.  NN0  <. (/) ,  E >. RegGrph  k )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753   _Vcvv 3058   (/)c0 3737   <.cop 3977   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   NN0cn0 10835   VDeg cvdg 25297   RegGrph crgra 25326
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-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-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-iota 5532  df-fv 5576  df-ov 6280  df-oprab 6281  df-rgra 25328
This theorem is referenced by: (None)
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