Theorem isofld 31174

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 31170	. 2 ⊢ oField = (Field ∩ oRing)
2	1	elin2 4097	1 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))

Syntax hints:   ↔ wb 209   ∧ wa 399   ∈ wcel 2112  Fieldcfield 19722  oRingcorng 31167  oFieldcofld 31168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-v 3400  df-in 3860  df-ofld 31170

This theorem is referenced by:  ofldfld  31182  ofldtos  31183  ofldlt1  31185  ofldchr  31186  subofld  31188  isarchiofld  31189  reofld  31212  nn0omnd  31213