Theorem iscrngo 38731

Description: Obsolete theorem, use iscrng 20378 instead. The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
iscrngo	⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))

Proof of Theorem iscrngo

Step	Hyp	Ref	Expression
1		df-crngo 38729	. 2 ⊢ CRingOps = (RingOps ∩ Com2)
2	1	elin2 4152	1 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2145  RingOpscrngo 38629  Com2ccm2 38724  CRingOpsccring 38728

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-in 3909  df-crngo 38729

This theorem is used by:  iscrngo2  38732  iscringd  38733  crngorngo  38735  fldcrngo  38739  isfld2  38740  isdmn2  38790