Theorem isofld 20783

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 20779	. 2 ⊢ oField = (Field ∩ oRing)
2	1	elin2 4152	1 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2113  Fieldcfield 20649  oRingcorng 20776  oFieldcofld 20777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-in 3905  df-ofld 20779

This theorem is referenced by:  ofldfld  20791  ofldtos  20792  ofldlt1  20794  subofld  20796  ofldchr  21517  isarchiofld  33177  reofld  33317  nn0omnd  33318