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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 ComplUSGrph USGrph
Distinct variable groups:   ,,   ,,

Proof of Theorem iscusgra0
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cusgra 24850 . . . . 5 ComplUSGrph USGrph
21relopabi 4950 . . . 4 ComplUSGrph
32brrelexi 4866 . . 3 ComplUSGrph
42brrelex2i 4867 . . 3 ComplUSGrph
53, 4jca 532 . 2 ComplUSGrph
6 iscusgra 24885 . . 3 ComplUSGrph USGrph
76biimpd 209 . 2 ComplUSGrph USGrph
85, 7mpcom 36 1 ComplUSGrph USGrph
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wcel 1844  wral 2756  cvv 3061   cdif 3413  csn 3974  cpr 3976   class class class wbr 4397   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
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