Theorem isofld 20900

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 20896	. 2 ⊢ oField = (Field ∩ oRing)
2	1	elin2 4153	1 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))

Syntax hints:   ↔ wb 208   ∧ wa 399   ∈ wcel 2141  Fieldcfield 20766  oRingcorng 20893  oFieldcofld 20894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-in 3909  df-ofld 20896

This theorem is referenced by:  ofldfld  20908  ofldtos  20909  ofldlt1  20911  subofld  20913  ofldchr  21615  isarchiofld  33339  reofld  33489  nn0omnd  33490