Theorem negpitopissre

Description: The interval ( -upi (,] pi ) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Assertion	negpitopissre	$$⊢\left(-\mathrm{\pi },\mathrm{\pi }\right]\subseteq ℝ$$

Proof

Step	Hyp	Ref	Expression
1		pire	$$⊢\mathrm{\pi }\in ℝ$$
2	1	renegcli	$$⊢-\mathrm{\pi }\in ℝ$$
3	2	rexri	$$⊢-\mathrm{\pi }\in {ℝ}^{*}$$
4		iocssre	$$⊢\left(-\mathrm{\pi }\in {ℝ}^{*}\wedge \mathrm{\pi }\in ℝ\right)\to \left(-\mathrm{\pi },\mathrm{\pi }\right]\subseteq ℝ$$
5	3 1 4	mp2an	$$⊢\left(-\mathrm{\pi },\mathrm{\pi }\right]\subseteq ℝ$$