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Theorem isofld 33180
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 33176 . 2 oField = (Field ∩ oRing)
21elin2 4198 1 (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  wcel 2099  Fieldcfield 20708  oRingcorng 33173  oFieldcofld 33174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464  df-in 3954  df-ofld 33176
This theorem is referenced by:  ofldfld  33188  ofldtos  33189  ofldlt1  33191  ofldchr  33192  subofld  33194  isarchiofld  33195  reofld  33219  nn0omnd  33220
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