Theorem isofld 20809

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 20805	. 2 ⊢ oField = (Field ∩ oRing)
2	1	elin2 4157	1 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  Fieldcfield 20675  oRingcorng 20802  oFieldcofld 20803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-in 3910  df-ofld 20805

This theorem is referenced by:  ofldfld  20817  ofldtos  20818  ofldlt1  20820  subofld  20822  ofldchr  21543  isarchiofld  33292  reofld  33435  nn0omnd  33436