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Theorem isprrngo 26662
 Description: The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
isprrng.1
isprrng.2 GId
Assertion
Ref Expression
isprrngo

Proof of Theorem isprrngo
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fveq2 5730 . . . . . . 7
2 isprrng.1 . . . . . . 7
31, 2syl6eqr 2488 . . . . . 6
43fveq2d 5734 . . . . 5 GId GId
5 isprrng.2 . . . . 5 GId
64, 5syl6eqr 2488 . . . 4 GId
76sneqd 3829 . . 3 GId
8 fveq2 5730 . . 3
97, 8eleq12d 2506 . 2 GId
10 df-prrngo 26660 . 2 GId
119, 10elrab2 3096 1
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   wceq 1653   wcel 1726  csn 3816  cfv 5456  c1st 6349  GIdcgi 21777  crngo 21965  cpridl 26620  cprrng 26658 This theorem is referenced by:  prrngorngo  26663  smprngopr  26664  isdmn3  26686 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-prrngo 26660
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