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Theorem isidom 18163
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 18143 . 2  |- IDomn  =  (
CRing  i^i Domn )
21elin2 3627 1  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    e. wcel 1840   CRingccrg 17409  Domncdomn 18138  IDomncidom 18139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-v 3058  df-in 3418  df-idom 18143
This theorem is referenced by:  fldidom  18164  fiidomfld  18167  znfld  18787  znidomb  18788  ply1idom  22707  fta1glem1  22748  fta1glem2  22749  fta1g  22750  fta1b  22752  lgsqrlem1  23887  lgsqrlem2  23888  lgsqrlem3  23889  lgsqrlem4  23890  idomrootle  35480  idomodle  35481  proot1mul  35484  proot1hash  35488
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