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

Step	Hyp	Ref	Expression
1		df-crngo 28967	. 2 |- CRingOps = ( RingOps i^i Com2 )
2	1	elin2 3652	1 |- ( R e. CRingOps <-> ( R e. RingOps /\ R e. Com2 ) )

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