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Theorem isdmn 26706
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 26701 . 2  |-  Dmn  =  ( PrRing  i^i  Com2 )
21elin2 3520 1  |-  ( R  e.  Dmn  <->  ( R  e.  PrRing  /\  R  e.  Com2 ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    e. wcel 1728   Com2ccm2 22036   PrRingcprrng 26698   Dmncdmn 26699
This theorem is referenced by:  isdmn2  26707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-v 2967  df-in 3316  df-dmn 26701
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