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Theorem iscrngo 38003
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 38001 . 2 CRingOps = (RingOps ∩ Com2)
21elin2 4203 1 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2108  RingOpscrngo 37901  Com2ccm2 37996  CRingOpsccring 38000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-in 3958  df-crngo 38001
This theorem is referenced by:  iscrngo2  38004  iscringd  38005  crngorngo  38007  fldcrngo  38011  isfld2  38012  isdmn2  38062
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