Theorem iscrngo 37168

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 37166	. 2 ⊢ CRingOps = (RingOps ∩ Com2)
2	1	elin2 4197	1 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∈ wcel 2105  RingOpscrngo 37066  Com2ccm2 37161  CRingOpsccring 37165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3955  df-crngo 37166

This theorem is referenced by:  iscrngo2  37169  iscringd  37170  crngorngo  37172  fldcrngo  37176  isfld2  37177  isdmn2  37227