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Theorem isprrngo 30609
 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 5872 . . . . . . 7
2 isprrng.1 . . . . . . 7
31, 2syl6eqr 2516 . . . . . 6
43fveq2d 5876 . . . . 5 GId GId
5 isprrng.2 . . . . 5 GId
64, 5syl6eqr 2516 . . . 4 GId
76sneqd 4044 . . 3 GId
8 fveq2 5872 . . 3
97, 8eleq12d 2539 . 2 GId
10 df-prrngo 30607 . 2 GId
119, 10elrab2 3259 1
 Colors of variables: wff setvar class Syntax hints:   wb 184   wa 369   wceq 1395   wcel 1819  csn 4032  cfv 5594  c1st 6797  GIdcgi 25315  crngo 25503  cpridl 30567  cprrng 30605 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-prrngo 30607 This theorem is referenced by:  prrngorngo  30610  smprngopr  30611  isdmn3  30633
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