Theorem iscrngo 38500

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 38498	. 2 ⊢ CRingOps = (RingOps ∩ Com2)
2	1	elin2 4157	1 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))

Syntax hints:   ↔ wb 208   ∧ wa 399   ∈ wcel 2144  RingOpscrngo 38398  Com2ccm2 38493  CRingOpsccring 38497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-v 3458  df-in 3913  df-crngo 38498

This theorem is referenced by:  iscrngo2  38501  iscringd  38502  crngorngo  38504  fldcrngo  38508  isfld2  38509  isdmn2  38559