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Theorem iscrngo 26588
Description: The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
Assertion
Ref Expression
iscrngo  |-  ( R  e. CRingOps 
<->  ( R  e.  RingOps  /\  R  e.  Com2 ) )

Proof of Theorem iscrngo
StepHypRef Expression
1 df-crngo 26587 . 2  |- CRingOps  =  (
RingOps  i^i  Com2 )
21elin2 3523 1  |-  ( R  e. CRingOps 
<->  ( R  e.  RingOps  /\  R  e.  Com2 ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1725   RingOpscrngo 21955   Com2ccm2 21990  CRingOpsccring 26586
This theorem is referenced by:  iscrngo2  26589  iscringd  26590  crngorngo  26591  fldcrng  26595  isfld2  26596  isdmn2  26646
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319  df-crngo 26587
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