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Theorem isfld 15771
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 15765 . 2  |- Field  =  (
DivRing  i^i  CRing )
21elin2 3474 1  |-  ( R  e. Field 
<->  ( R  e.  DivRing  /\  R  e.  CRing ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1717   CRingccrg 15588   DivRingcdr 15762  Fieldcfield 15763
This theorem is referenced by:  fldpropd  15790  fldidom  16292  fiidomfld  16295  ply1pid  19969  lgseisenlem3  21002  lgseisenlem4  21003  ofldsqr  24066  ofldaddlt  24067  ofld0le1  24068  ofldlt1  24069  ofldchr  24070  subofld  24071  refld  24095  qqhrhm  24172  rrhre  24183
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 2368
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 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-in 3270  df-field 15765
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