Theorem isofld 31501

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 31497	. 2 ⊢ oField = (Field ∩ oRing)
2	1	elin2 4131	1 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  Fieldcfield 19992  oRingcorng 31494  oFieldcofld 31495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ofld 31497

This theorem is referenced by:  ofldfld  31509  ofldtos  31510  ofldlt1  31512  ofldchr  31513  subofld  31515  isarchiofld  31516  reofld  31544  nn0omnd  31545