Theorem iscrngo 38931

Description: Obsolete theorem, use iscrng 20466 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 38929	. 2 ⊢ CRingOps = (RingOps ∩ Com2)
2	1	elin2 4100	1 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2144  RingOpscrngo 38829  Com2ccm2 38924  CRingOpsccring 38928

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-in 4091  df-crngo 38929

This theorem is used by:  iscrngo2  38932  iscringd  38933  crngorngo  38935  fldcrngo  38939  isfld2  38940  isdmn2  38990