Theorem iscrngo 31657

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 31656	. 2 |- CRingOps = ( RingOps i^i Com2 )
2	1	elin2 3629	1 |- ( R e. CRingOps <-> ( R e. RingOps /\ R e. Com2 ) )

Syntax hints:   <-> wb 184   /\ wa 367   e. wcel 1842   RingOpscrngo 25671   Com2ccm2 25706  CRingOpsccring 31655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380

This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3060  df-in 3420  df-crngo 31656

This theorem is referenced by:  iscrngo2  31658  iscringd  31659  crngorngo  31660  fldcrng  31664  isfld2  31665  isdmn2  31715