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Theorem iscrngo 32230
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 32229 . 2  |- CRingOps  =  (
RingOps  i^i  Com2 )
21elin2 3621 1  |-  ( R  e. CRingOps 
<->  ( R  e.  RingOps  /\  R  e.  Com2 ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    e. wcel 1887   RingOpscrngo 26103   Com2ccm2 26138  CRingOpsccring 32228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-in 3411  df-crngo 32229
This theorem is referenced by:  iscrngo2  32231  iscringd  32232  crngorngo  32233  fldcrng  32237  isfld2  32238  isdmn2  32288
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