Theorem iscrngo 33795

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 33793	. 2 |- CRingOps = ( RingOps i^i Com2 )
2	1	elin2 3801	1 |- ( R e. CRingOps <-> ( R e. RingOps /\ R e. Com2 ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   RingOpscrngo 33693   Com2ccm2 33788  CRingOpsccring 33792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-crngo 33793

This theorem is referenced by:  iscrngo2  33796  iscringd  33797  crngorngo  33799  fldcrng  33803  isfld2  33804  isdmn2  33854