Theorem iscrngo 39121

Description: Obsolete theorem, use iscrng 20493 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 39119	. 2 ⊢ CRingOps = (RingOps ∩ Com2)
2	1	elin2 4103	1 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2147  RingOpscrngo 39019  Com2ccm2 39114  CRingOpsccring 39118

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-in 4094  df-crngo 39119

This theorem is used by:  iscrngo2  39122  iscringd  39123  crngorngo  39125  fldcrngo  39129  isfld2  39130  isdmn2  39180