Theorem iscrngo 37983

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 37981	. 2 ⊢ CRingOps = (RingOps ∩ Com2)
2	1	elin2 4213	1 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2106  RingOpscrngo 37881  Com2ccm2 37976  CRingOpsccring 37980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-in 3970  df-crngo 37981

This theorem is referenced by:  iscrngo2  37984  iscringd  37985  crngorngo  37987  fldcrngo  37991  isfld2  37992  isdmn2  38042