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Theorem iscusgra0 23502
Description: The property of being a complete (undirected simple) graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
Assertion
Ref Expression
iscusgra0  |-  ( V ComplUSGrph  E  ->  ( V USGrph  E  /\  A. k  e.  V  A. n  e.  ( V  \  { k } ) { n ,  k }  e.  ran  E ) )
Distinct variable groups:    k, n, E    k, V, n

Proof of Theorem iscusgra0
Dummy variables  e 
v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cusgra 23470 . . . . 5  |- ComplUSGrph  =  { <. v ,  e >.  |  ( v USGrph  e  /\  A. k  e.  v 
A. n  e.  ( v  \  { k } ) { n ,  k }  e.  ran  e ) }
21relopabi 5065 . . . 4  |-  Rel ComplUSGrph
32brrelexi 4979 . . 3  |-  ( V ComplUSGrph  E  ->  V  e.  _V )
42brrelex2i 4980 . . 3  |-  ( V ComplUSGrph  E  ->  E  e.  _V )
53, 4jca 532 . 2  |-  ( V ComplUSGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
6 iscusgra 23501 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V ComplUSGrph  E  <->  ( V USGrph  E  /\  A. k  e.  V  A. n  e.  ( V  \  {
k } ) { n ,  k }  e.  ran  E ) ) )
76biimpd 207 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V ComplUSGrph  E  ->  ( V USGrph  E  /\  A. k  e.  V  A. n  e.  ( V  \  {
k } ) { n ,  k }  e.  ran  E ) ) )
85, 7mpcom 36 1  |-  ( V ComplUSGrph  E  ->  ( V USGrph  E  /\  A. k  e.  V  A. n  e.  ( V  \  { k } ) { n ,  k }  e.  ran  E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1758   A.wral 2795   _Vcvv 3070    \ cdif 3425   {csn 3977   {cpr 3979   class class class wbr 4392   ran crn 4941   USGrph cusg 23401   ComplUSGrph ccusgra 23467
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 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
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-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-br 4393  df-opab 4451  df-xp 4946  df-rel 4947  df-cnv 4948  df-dm 4950  df-rn 4951  df-cusgra 23470
This theorem is referenced by:  cusisusgra  23503  cusgrarn  23504  cusgrares  23517  usgrasscusgra  23528  sizeusglecusglem1  23529
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