Theorem isofld 31497

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 31493	. 2 ⊢ oField = (Field ∩ oRing)
2	1	elin2 4136	1 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∈ wcel 2110  Fieldcfield 19990  oRingcorng 31490  oFieldcofld 31491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-v 3433  df-in 3899  df-ofld 31493

This theorem is referenced by:  ofldfld  31505  ofldtos  31506  ofldlt1  31508  ofldchr  31509  subofld  31511  isarchiofld  31512  reofld  31540  nn0omnd  31541