Theorem isofld 21127

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 21123	. 2 ⊢ oField = (Field ∩ oRing)
2	1	elin2 4100	1 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2144  Fieldcfield 20987  oRingcorng 21120  oFieldcofld 21121

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-in 4091  df-ofld 21123

This theorem is used by:  ofldfld  21135  ofldtos  21136  ofldlt1  21138  subofld  21140  ofldchr  21888  isarchiofld  33807  reofld  33951  nn0omnd  33952