Theorem ofldfld 21134

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 21126	. 2 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
2	1	simplbi 502	1 ⊢ (𝐹 ∈ oField → 𝐹 ∈ Field)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  Fieldcfield 20986  oRingcorng 21119  oFieldcofld 21120

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 21122

This theorem is used by:  ofldlt1  21137  ofldchr  21887