Theorem isofld 33333

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 33329	. 2 ⊢ oField = (Field ∩ oRing)
2	1	elin2 4202	1 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2107  Fieldcfield 20731  oRingcorng 33326  oFieldcofld 33327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-in 3957  df-ofld 33329

This theorem is referenced by:  ofldfld  33341  ofldtos  33342  ofldlt1  33344  ofldchr  33345  subofld  33347  isarchiofld  33348  reofld  33373  nn0omnd  33374