Theorem iscrngo 38199

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 38197	. 2 ⊢ CRingOps = (RingOps ∩ Com2)
2	1	elin2 4156	1 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  RingOpscrngo 38097  Com2ccm2 38192  CRingOpsccring 38196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3443  df-in 3909  df-crngo 38197

This theorem is referenced by:  iscrngo2  38200  iscringd  38201  crngorngo  38203  fldcrngo  38207  isfld2  38208  isdmn2  38258