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Theorem isdmn 26356
Description: The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
Assertion
Ref Expression
isdmn  |-  ( R  e.  Dmn  <->  ( R  e.  PrRing  /\  R  e.  Com2 ) )

Proof of Theorem isdmn
StepHypRef Expression
1 df-dmn 26351 . 2  |-  Dmn  =  ( PrRing  i^i  Com2 )
21elin2 3475 1  |-  ( R  e.  Dmn  <->  ( R  e.  PrRing  /\  R  e.  Com2 ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1717   Com2ccm2 21847   PrRingcprrng 26348   Dmncdmn 26349
This theorem is referenced by:  isdmn2  26357
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-v 2902  df-in 3271  df-dmn 26351
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