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Theorem isofld 30929
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 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))

Proof of Theorem isofld
StepHypRef Expression
1 df-ofld 30925 . 2 oField = (Field ∩ oRing)
21elin2 4127 1 (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wcel 2112  Fieldcfield 19499  oRingcorng 30922  oFieldcofld 30923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-in 3891  df-ofld 30925
This theorem is referenced by:  ofldfld  30937  ofldtos  30938  ofldlt1  30940  ofldchr  30941  subofld  30943  isarchiofld  30944  reofld  30967  nn0omnd  30968
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