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

Step	Hyp	Ref	Expression
1		df-crngo 32229	. 2 |- CRingOps = ( RingOps i^i Com2 )
2	1	elin2 3621	1 |- ( R e. CRingOps <-> ( R e. RingOps /\ R e. Com2 ) )

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