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Theorem isidom 16094
Description: An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
Assertion
Ref Expression
isidom  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )

Proof of Theorem isidom
StepHypRef Expression
1 df-idom 16075 . 2  |- IDomn  =  (
CRing  i^i Domn )
21elin2 3393 1  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1701   CRingccrg 15387  Domncdomn 16070  IDomncidom 16071
This theorem is referenced by:  fldidom  16095  fiidomfld  16098  znfld  16570  znidomb  16571  ply1idom  19563  fta1glem1  19604  fta1glem2  19605  fta1g  19606  fta1b  19608  lgsqrlem1  20633  lgsqrlem2  20634  lgsqrlem3  20635  lgsqrlem4  20636  idomrootle  26659  idomodle  26660  proot1mul  26663  proot1hash  26667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-v 2824  df-in 3193  df-idom 16075
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