Theorem iscrngo 38378

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 38376	. 2 ⊢ CRingOps = (RingOps ∩ Com2)
2	1	elin2 4135	1 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))

Syntax hints:   ↔ wb 208   ∧ wa 397   ∈ wcel 2121  RingOpscrngo 38276  Com2ccm2 38371  CRingOpsccring 38375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-v 3435  df-in 3892  df-crngo 38376

This theorem is referenced by:  iscrngo2  38379  iscringd  38380  crngorngo  38382  fldcrngo  38386  isfld2  38387  isdmn2  38437