Theorem ofldfld 20817

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 20809	. 2 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
2	1	simplbi 496	1 ⊢ (𝐹 ∈ oField → 𝐹 ∈ Field)

Syntax hints:   → wi 4   ∈ wcel 2114  Fieldcfield 20675  oRingcorng 20802  oFieldcofld 20803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-in 3910  df-ofld 20805

This theorem is referenced by:  ofldlt1  20820  ofldchr  21543