Theorem isofld 33180

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 33176	. 2 ⊢ oField = (Field ∩ oRing)
2	1	elin2 4198	1 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))

Syntax hints:   ↔ wb 205   ∧ wa 394   ∈ wcel 2099  Fieldcfield 20708  oRingcorng 33173  oFieldcofld 33174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464  df-in 3954  df-ofld 33176

This theorem is referenced by:  ofldfld  33188  ofldtos  33189  ofldlt1  33191  ofldchr  33192  subofld  33194  isarchiofld  33195  reofld  33219  nn0omnd  33220