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Theorem iscrngo 26725
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 26724 . 2  |- CRingOps  =  (
RingOps  i^i  Com2 )
21elin2 3372 1  |-  ( R  e. CRingOps 
<->  ( R  e.  RingOps  /\  R  e.  Com2 ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1696   RingOpscrngo 21058   Com2ccm2 21093  CRingOpsccring 26723
This theorem is referenced by:  iscrngo2  26726  iscringd  26727  crngorngo  26728  fldcrng  26732  isfld2  26733  isdmn2  26783
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-crngo 26724
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