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Theorem isofld 29611
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 29607 . 2 oField = (Field ∩ oRing)
21elin2 3784 1 (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wcel 1987  Fieldcfield 18680  oRingcorng 29604  oFieldcofld 29605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-in 3566  df-ofld 29607
This theorem is referenced by:  ofldfld  29619  ofldtos  29620  ofldlt1  29622  ofldchr  29623  subofld  29625  isarchiofld  29626  reofld  29649  nn0omnd  29650
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