Theorem isofld 20933

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

Step	Hyp	Ref	Expression
1		df-ofld 20929	. 2 ⊢ oField = (Field ∩ oRing)
2	1	elin2 4158	1 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∈ wcel 2145  Fieldcfield 20802  oRingcorng 20926  oFieldcofld 20927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-in 3914  df-ofld 20929

This theorem is referenced by:  ofldfld  20941  ofldtos  20942  ofldlt1  20944  subofld  20946  ofldchr  21683  isarchiofld  33427  reofld  33573  nn0omnd  33574