Theorem xrconn

Description: The topology of the extended reals is connected. (Contributed by Mario Carneiro, 9-Sep-2015)

	Ref	Expression
Assertion	xrconn	$$⊢\mathrm{ordTop}\left(\le \right)\in \mathrm{Conn}$$

Proof

Step	Hyp	Ref	Expression
1		xrhmph	$$⊢\mathrm{II}\simeq \mathrm{ordTop}\left(\le \right)$$
2		iiconn	$$⊢\mathrm{II}\in \mathrm{Conn}$$
3		connhmph	$$⊢\mathrm{II}\simeq \mathrm{ordTop}\left(\le \right)\to \left(\mathrm{II}\in \mathrm{Conn}\to \mathrm{ordTop}\left(\le \right)\in \mathrm{Conn}\right)$$
4	1 2 3	mp2	$$⊢\mathrm{ordTop}\left(\le \right)\in \mathrm{Conn}$$