Theorem isofld 21004

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 21000	. 2 ⊢ oField = (Field ∩ oRing)
2	1	elin2 4159	1 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2146  Fieldcfield 20865  oRingcorng 20997  oFieldcofld 20998

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-in 3915  df-ofld 21000

This theorem is used by:  ofldfld  21012  ofldtos  21013  ofldlt1  21015  subofld  21017  ofldchr  21763  isarchiofld  33550  reofld  33694  nn0omnd  33695