Theorem ofldfld 31099

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 31091	. 2 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
2	1	simplbi 501	1 ⊢ (𝐹 ∈ oField → 𝐹 ∈ Field)

Syntax hints:   → wi 4   ∈ wcel 2114  Fieldcfield 19635  oRingcorng 31084  oFieldcofld 31085

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 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-v 3402  df-in 3860  df-ofld 31087

This theorem is referenced by:  ofldlt1  31102  ofldchr  31103