Theorem isofld 20948

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 20944	. 2 ⊢ oField = (Field ∩ oRing)
2	1	elin2 4157	1 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∈ wcel 2143  Fieldcfield 20815  oRingcorng 20941  oFieldcofld 20942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-in 3913  df-ofld 20944

This theorem is referenced by:  ofldfld  20956  ofldtos  20957  ofldlt1  20959  subofld  20961  ofldchr  21707  isarchiofld  33500  reofld  33644  nn0omnd  33645