Theorem isofld 32690

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 32686	. 2 ⊢ oField = (Field ∩ oRing)
2	1	elin2 4196	1 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))

Syntax hints:   ↔ wb 205   ∧ wa 394   ∈ wcel 2104  Fieldcfield 20501  oRingcorng 32683  oFieldcofld 32684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-in 3954  df-ofld 32686

This theorem is referenced by:  ofldfld  32698  ofldtos  32699  ofldlt1  32701  ofldchr  32702  subofld  32704  isarchiofld  32705  reofld  32729  nn0omnd  32730