MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscusgra0 Structured version   Unicode version

Theorem iscusgra0 24886
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 24850 . . . . 5  |- ComplUSGrph  =  { <. v ,  e >.  |  ( v USGrph  e  /\  A. k  e.  v 
A. n  e.  ( v  \  { k } ) { n ,  k }  e.  ran  e ) }
21relopabi 4950 . . . 4  |-  Rel ComplUSGrph
32brrelexi 4866 . . 3  |-  ( V ComplUSGrph  E  ->  V  e.  _V )
42brrelex2i 4867 . . 3  |-  ( V ComplUSGrph  E  ->  E  e.  _V )
53, 4jca 532 . 2  |-  ( V ComplUSGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
6 iscusgra 24885 . . 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 209 . 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 1844   A.wral 2756   _Vcvv 3061    \ cdif 3413   {csn 3974   {cpr 3976   class class class wbr 4397   ran crn 4826   USGrph cusg 24759   ComplUSGrph ccusgra 24847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-br 4398  df-opab 4456  df-xp 4831  df-rel 4832  df-cnv 4833  df-dm 4835  df-rn 4836  df-cusgra 24850
This theorem is referenced by:  cusisusgra  24887  cusgrarn  24888  cusgrares  24901  usgrasscusgra  24912  sizeusglecusglem1  24913
  Copyright terms: Public domain W3C validator