MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0vgrargra Structured version   Unicode version

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  |-  ( 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 462 . . 3  |-  ( ( E  e.  V  /\  k  e.  NN0 )  -> 
k  e.  NN0 )
2 ral0 3899 . . . 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 4549 . . . 4  |-  (/)  e.  _V
5 isrgra 25630 . . . 4  |-  ( (
(/)  e.  _V  /\  E  e.  V  /\  k  e.  NN0 )  ->  ( <.
(/) ,  E >. RegGrph  k  <-> 
( k  e.  NN0  /\ 
A. p  e.  (/)  ( ( (/) VDeg  E ) `
 p )  =  k ) ) )
64, 5mp3an1 1347 . . 3  |-  ( ( E  e.  V  /\  k  e.  NN0 )  -> 
( <. (/) ,  E >. RegGrph  k  <-> 
( k  e.  NN0  /\ 
A. p  e.  (/)  ( ( (/) VDeg  E ) `
 p )  =  k ) ) )
71, 3, 6mpbir2and 930 . 2  |-  ( ( E  e.  V  /\  k  e.  NN0 )  ->  <.
(/) ,  E >. RegGrph  k )
87ralrimiva 2837 1  |-  ( E  e.  V  ->  A. k  e.  NN0  <. (/) ,  E >. RegGrph  k )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867   A.wral 2773   _Vcvv 3078   (/)c0 3758   <.cop 3999   class class class wbr 4417   ` cfv 5593  (class class class)co 6297   NN0cn0 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)
  Copyright terms: Public domain W3C validator