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Theorem iscrngo 35266
 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 35264 . 2 CRingOps = (RingOps ∩ Com2)
21elin2 4172 1 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   ∧ wa 398   ∈ wcel 2107  RingOpscrngo 35164  Com2ccm2 35259  CRingOpsccring 35263 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-in 3941  df-crngo 35264 This theorem is referenced by:  iscrngo2  35267  iscringd  35268  crngorngo  35270  fldcrng  35274  isfld2  35275  isdmn2  35325
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