Theorem ofldfld 20851

Description: An ordered field is a field. (Contributed by Thierry Arnoux, 20-Jan-2018.)

Assertion

Ref	Expression
ofldfld	⊢ (𝐹 ∈ oField → 𝐹 ∈ Field)

Proof of Theorem ofldfld

Step	Hyp	Ref	Expression
1		isofld 20843	. 2 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
2	1	simplbi 497	1 ⊢ (𝐹 ∈ oField → 𝐹 ∈ Field)

Syntax hints:   → wi 4   ∈ wcel 2119  Fieldcfield 20709  oRingcorng 20836  oFieldcofld 20837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-in 3897  df-ofld 20839

This theorem is referenced by:  ofldlt1  20854  ofldchr  21558