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Theorem iscrngo 28968
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 28967 . 2  |- CRingOps  =  (
RingOps  i^i  Com2 )
21elin2 3652 1  |-  ( R  e. CRingOps 
<->  ( R  e.  RingOps  /\  R  e.  Com2 ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    e. wcel 1758   RingOpscrngo 24041   Com2ccm2 24076  CRingOpsccring 28966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-in 3446  df-crngo 28967
This theorem is referenced by:  iscrngo2  28969  iscringd  28970  crngorngo  28971  fldcrng  28975  isfld2  28976  isdmn2  29026
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