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Theorem iscrngo 26298
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 26297 . 2  |- CRingOps  =  (
RingOps  i^i  Com2 )
21elin2 3474 1  |-  ( R  e. CRingOps 
<->  ( R  e.  RingOps  /\  R  e.  Com2 ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1717   RingOpscrngo 21811   Com2ccm2 21846  CRingOpsccring 26296
This theorem is referenced by:  iscrngo2  26299  iscringd  26300  crngorngo  26301  fldcrng  26305  isfld2  26306  isdmn2  26356
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-in 3270  df-crngo 26297
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