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Theorem iscrngo 35896
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
StepHypRef Expression
1 df-crngo 35894 . 2 CRingOps = (RingOps ∩ Com2)
21elin2 4116 1 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wcel 2110  RingOpscrngo 35794  Com2ccm2 35889  CRingOpsccring 35893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3415  df-in 3878  df-crngo 35894
This theorem is referenced by:  iscrngo2  35897  iscringd  35898  crngorngo  35900  fldcrng  35904  isfld2  35905  isdmn2  35955
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