Theorem iscrngo 38020

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 38018	. 2 ⊢ CRingOps = (RingOps ∩ Com2)
2	1	elin2 4178	1 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  RingOpscrngo 37918  Com2ccm2 38013  CRingOpsccring 38017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-in 3933  df-crngo 38018

This theorem is referenced by:  iscrngo2  38021  iscringd  38022  crngorngo  38024  fldcrngo  38028  isfld2  38029  isdmn2  38079