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Theorem iscusgra0 24133
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 24097 . . . . 5  |- ComplUSGrph  =  { <. v ,  e >.  |  ( v USGrph  e  /\  A. k  e.  v 
A. n  e.  ( v  \  { k } ) { n ,  k }  e.  ran  e ) }
21relopabi 5126 . . . 4  |-  Rel ComplUSGrph
32brrelexi 5039 . . 3  |-  ( V ComplUSGrph  E  ->  V  e.  _V )
42brrelex2i 5040 . . 3  |-  ( V ComplUSGrph  E  ->  E  e.  _V )
53, 4jca 532 . 2  |-  ( V ComplUSGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
6 iscusgra 24132 . . 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 1767   A.wral 2814   _Vcvv 3113    \ cdif 3473   {csn 4027   {cpr 4029   class class class wbr 4447   ran crn 5000   USGrph cusg 24006   ComplUSGrph ccusgra 24094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-dm 5009  df-rn 5010  df-cusgra 24097
This theorem is referenced by:  cusisusgra  24134  cusgrarn  24135  cusgrares  24148  usgrasscusgra  24159  sizeusglecusglem1  24160
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