Theorem isofld 31403

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 31399	. 2 ⊢ oField = (Field ∩ oRing)
2	1	elin2 4127	1 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∈ wcel 2108  Fieldcfield 19907  oRingcorng 31396  oFieldcofld 31397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-ofld 31399

This theorem is referenced by:  ofldfld  31411  ofldtos  31412  ofldlt1  31414  ofldchr  31415  subofld  31417  isarchiofld  31418  reofld  31446  nn0omnd  31447