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Theorem isofld 26401
Description: An ordered field is a field with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 23-Mar-2018.)
Assertion
Ref Expression
isofld  |-  ( F  e. oField 
<->  ( F  e. Field  /\  F  e. oRing ) )

Proof of Theorem isofld
StepHypRef Expression
1 df-ofld 26397 . 2  |- oField  =  (Field 
i^i oRing )
21elin2 3636 1  |-  ( F  e. oField 
<->  ( F  e. Field  /\  F  e. oRing ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    e. wcel 1758  Fieldcfield 16936  oRingcorng 26394  oFieldcofld 26395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-v 3067  df-in 3430  df-ofld 26397
This theorem is referenced by:  ofldfld  26409  ofldtos  26410  ofldlt1  26412  ofldchr  26413  subofld  26415  isarchiofld  26416  reofld  26439  nn0omnd  26440
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