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Theorem isusgra 25064
 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 5772 . . . 4
3 dmeq 5034 . . . . 5
43adantl 468 . . . 4
5 f1eq2 5773 . . . 4
64, 5syl 17 . . 3
7 simpl 459 . . . . . 6
87pweqd 3955 . . . . 5
98difeq1d 3549 . . . 4
10 rabeq 3037 . . . 4
11 f1eq3 5774 . . . 4
129, 10, 113syl 18 . . 3
132, 6, 123bitrd 283 . 2
14 df-usgra 25053 . 2 USGrph
1513, 14brabga 4714 1 USGrph
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 188   wa 371   wceq 1443   wcel 1886  crab 2740   cdif 3400  c0 3730  cpw 3950  csn 3967   class class class wbr 4401   cdm 4833  wf1 5578  cfv 5581  c2 10656  chash 12512   USGrph cusg 25050 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-rab 2745  df-v 3046  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-br 4402  df-opab 4461  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-usgra 25053 This theorem is referenced by:  usgraf  25066  isusgra0  25067  usgraeq12d  25082  usisuslgra  25085  usgrares  25089  usgra0  25090  usgra0v  25091  usgra1v  25110
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