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Theorem isfld 15836
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 15830 . 2  |- Field  =  (
DivRing  i^i  CRing )
21elin2 3523 1  |-  ( R  e. Field 
<->  ( R  e.  DivRing  /\  R  e.  CRing ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1725   CRingccrg 15653   DivRingcdr 15827  Fieldcfield 15828
This theorem is referenced by:  fldpropd  15855  fldidom  16357  fiidomfld  16360  ply1pid  20094  lgseisenlem3  21127  lgseisenlem4  21128  ofldsqr  24232  ofldaddlt  24233  ofld0le1  24234  ofldlt1  24235  ofldchr  24236  subofld  24237  refld  24271  qqhrhm  24365  rrhre  24379  sitmcl  24655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319  df-field 15830
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