Theorem xrsup

Description: The extended real numbers are unbounded above. (Contributed by Mario Carneiro, 7-May-2016)

	Ref	Expression
Assertion	xrsup	$$⊢sup\left({ℝ}^{*},{ℝ}^{*},<\right)=\mathrm{+\infty }$$

Proof

Step	Hyp	Ref	Expression
1		ssid	$$⊢{ℝ}^{*}\subseteq {ℝ}^{*}$$
2		pnfxr	$$⊢\mathrm{+\infty }\in {ℝ}^{*}$$
3		supxrpnf	$$⊢\left({ℝ}^{*}\subseteq {ℝ}^{*}\wedge \mathrm{+\infty }\in {ℝ}^{*}\right)\to sup\left({ℝ}^{*},{ℝ}^{*},<\right)=\mathrm{+\infty }$$
4	1 2 3	mp2an	$$⊢sup\left({ℝ}^{*},{ℝ}^{*},<\right)=\mathrm{+\infty }$$