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Metamath Proof Explorer

Theorem ofldtos

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

		Ref	Expression
	Assertion	ofldtos	$⊢ F \in oField \to F \in Toset$

Step	Hyp	Ref	Expression
1		isofld	$⊢ F \in oField \leftrightarrow (F \in Field \land F \in oRing)$
2	1	simprbi	$⊢ F \in oField \to F \in oRing$
3		orngogrp	$⊢ F \in oRing \to F \in oGrp$
4		isogrp	$⊢ F \in oGrp \leftrightarrow (F \in Grp \land F \in oMnd)$
5	4	simprbi	$⊢ F \in oGrp \to F \in oMnd$
6		omndtos	$⊢ F \in oMnd \to F \in Toset$
7	2 3 5 6	4syl	$⊢ F \in oField \to F \in Toset$