Theorem iscrngo 38675

Description: Obsolete theorem, use iscrng 20326 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 38673	. 2 ⊢ CRingOps = (RingOps ∩ Com2)
2	1	elin2 4156	1 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   ∈ wcel 2143  RingOpscrngo 38573  Com2ccm2 38668  CRingOpsccring 38672

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-in 3912  df-crngo 38673

This theorem is used by:  iscrngo2  38676  iscringd  38677  crngorngo  38679  fldcrngo  38683  isfld2  38684  isdmn2  38734