Theorem ofldtos 33289

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

Assertion

Ref	Expression
ofldtos	⊢ (𝐹 ∈ oField → 𝐹 ∈ Toset)

Proof of Theorem ofldtos

Step	Hyp	Ref	Expression
1		isofld 33280	. . 3 ⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
2	1	simprbi 496	. 2 ⊢ (𝐹 ∈ oField → 𝐹 ∈ oRing)
3		orngogrp 33279	. 2 ⊢ (𝐹 ∈ oRing → 𝐹 ∈ oGrp)
4		isogrp 33016	. . 3 ⊢ (𝐹 ∈ oGrp ↔ (𝐹 ∈ Grp ∧ 𝐹 ∈ oMnd))
5	4	simprbi 496	. 2 ⊢ (𝐹 ∈ oGrp → 𝐹 ∈ oMnd)
6		omndtos 33019	. 2 ⊢ (𝐹 ∈ oMnd → 𝐹 ∈ Toset)
7	2, 3, 5, 6	4syl 19	1 ⊢ (𝐹 ∈ oField → 𝐹 ∈ Toset)

Syntax hints:   → wi 4   ∈ wcel 2109  Tosetctos 18375  Grpcgrp 18865  Fieldcfield 20639  oMndcomnd 33011  oGrpcogrp 33012  oRingcorng 33273  oFieldcofld 33274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390  df-omnd 33013  df-ogrp 33014  df-orng 33275  df-ofld 33276

This theorem is referenced by:  ofldchr  33292