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Theorem iscrngo 34123
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 34121 . 2 CRingOps = (RingOps ∩ Com2)
21elin2 3952 1 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  wcel 2145  RingOpscrngo 34021  Com2ccm2 34116  CRingOpsccring 34120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3730  df-crngo 34121
This theorem is referenced by:  iscrngo2  34124  iscringd  34125  crngorngo  34127  fldcrng  34131  isfld2  34132  isdmn2  34182
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