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Theorem isfld 15521
Description: A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
Assertion
Ref Expression
isfld  |-  ( R  e. Field 
<->  ( R  e.  DivRing  /\  R  e.  CRing ) )

Proof of Theorem isfld
StepHypRef Expression
1 df-field 15515 . 2  |- Field  =  (
DivRing  i^i  CRing )
21elin2 3359 1  |-  ( R  e. Field 
<->  ( R  e.  DivRing  /\  R  e.  CRing ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1684   CRingccrg 15338   DivRingcdr 15512  Fieldcfield 15513
This theorem is referenced by:  fldpropd  15540  fldidom  16046  fiidomfld  16049  ply1pid  19565  lgseisenlem3  20590  lgseisenlem4  20591
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-field 15515
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