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Theorem isusgra 21365
 Description: The property of being an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
Assertion
Ref Expression
isusgra USGrph
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem isusgra
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1eq1 5626 . . . 4
3 dmeq 5062 . . . . 5
43adantl 453 . . . 4
5 f1eq2 5627 . . . 4
64, 5syl 16 . . 3
7 simpl 444 . . . . . 6
87pweqd 3796 . . . . 5
98difeq1d 3456 . . . 4
10 rabeq 2942 . . . 4
11 f1eq3 5628 . . . 4
129, 10, 113syl 19 . . 3
132, 6, 123bitrd 271 . 2
14 df-usgra 21359 . 2 USGrph
1513, 14brabga 4461 1 USGrph
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  crab 2701   cdif 3309  c0 3620  cpw 3791  csn 3806   class class class wbr 4204   cdm 4870  wf1 5443  cfv 5446  c2 10041  chash 11610   USGrph cusg 21357 This theorem is referenced by:  usgraf  21367  isusgra0  21368  usgraeq12d  21377  usisuslgra  21379  usgrares  21381  usgra0  21382  usgra0v  21383  usgra1v  21401 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-usgra 21359
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