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Theorem iscusgra0 23187
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 23155 . . . . 5  |- ComplUSGrph  =  { <. v ,  e >.  |  ( v USGrph  e  /\  A. k  e.  v 
A. n  e.  ( v  \  { k } ) { n ,  k }  e.  ran  e ) }
21relopabi 4952 . . . 4  |-  Rel ComplUSGrph
32brrelexi 4866 . . 3  |-  ( V ComplUSGrph  E  ->  V  e.  _V )
42brrelex2i 4867 . . 3  |-  ( V ComplUSGrph  E  ->  E  e.  _V )
53, 4jca 529 . 2  |-  ( V ComplUSGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
6 iscusgra 23186 . . 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 1755   A.wral 2705   _Vcvv 2962    \ cdif 3313   {csn 3865   {cpr 3867   class class class wbr 4280   ran crn 4828   USGrph cusg 23086   ComplUSGrph ccusgra 23152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-br 4281  df-opab 4339  df-xp 4833  df-rel 4834  df-cnv 4835  df-dm 4837  df-rn 4838  df-cusgra 23155
This theorem is referenced by:  cusisusgra  23188  cusgrarn  23189  cusgrares  23202  usgrasscusgra  23213  sizeusglecusglem1  23214
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