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Theorem isofld 29133
 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 29129 . 2 oField = (Field ∩ oRing)
21elin2 3763 1 (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  Fieldcfield 18571  oRingcorng 29126  oFieldcofld 29127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ofld 29129 This theorem is referenced by:  ofldfld  29141  ofldtos  29142  ofldlt1  29144  ofldchr  29145  subofld  29147  isarchiofld  29148  reofld  29171  nn0omnd  29172
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