Theorem iscrngo 36081

Description: The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)

Assertion

Ref	Expression
iscrngo	⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))

Proof of Theorem iscrngo

Step	Hyp	Ref	Expression
1		df-crngo 36079	. 2 ⊢ CRingOps = (RingOps ∩ Com2)
2	1	elin2 4127	1 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∈ wcel 2108  RingOpscrngo 35979  Com2ccm2 36074  CRingOpsccring 36078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-crngo 36079

This theorem is referenced by:  iscrngo2  36082  iscringd  36083  crngorngo  36085  fldcrng  36089  isfld2  36090  isdmn2  36140