Theorem ofldfld 20789

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 20781	. 2 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
2	1	simplbi 497	1 ⊢ (𝐹 ∈ oField → 𝐹 ∈ Field)

Syntax hints:   → wi 4   ∈ wcel 2113  Fieldcfield 20647  oRingcorng 20774  oFieldcofld 20775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-in 3905  df-ofld 20777

This theorem is referenced by:  ofldlt1  20792  ofldchr  21515