Theorem relt

Description: The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018)

	Ref	Expression
Assertion	relt	$$⊢<={<}_{{ℝ}_{\mathrm{fld}}}$$

Proof

Step	Hyp	Ref	Expression
1		dflt2	$$⊢<=\le \setminus \mathrm{I}$$
2		df-refld	$$⊢{ℝ}_{\mathrm{fld}}={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}ℝ$$
3	2	ovexi	$$⊢{ℝ}_{\mathrm{fld}}\in \mathrm{V}$$
4		rele2	$$⊢\le ={\le }_{{ℝ}_{\mathrm{fld}}}$$
5		eqid	$$⊢{<}_{{ℝ}_{\mathrm{fld}}}={<}_{{ℝ}_{\mathrm{fld}}}$$
6	4 5	pltfval	$$⊢{ℝ}_{\mathrm{fld}}\in \mathrm{V}\to {<}_{{ℝ}_{\mathrm{fld}}}=\le \setminus \mathrm{I}$$
7	3 6	ax-mp	$$⊢{<}_{{ℝ}_{\mathrm{fld}}}=\le \setminus \mathrm{I}$$
8	1 7	eqtr4i	$$⊢<={<}_{{ℝ}_{\mathrm{fld}}}$$