Theorem icossre

Description: A closed-below interval with real lower bound is a set of reals. (Contributed by Mario Carneiro, 14-Jun-2014)

	Ref	Expression
Assertion	icossre	|- ( ( A e. RR /\ B e. RR* ) -> ( A [,) B ) C_ RR )

Proof

Step	Hyp	Ref	Expression
1		elico2	|- ( ( A e. RR /\ B e. RR* ) -> ( x e. ( A [,) B ) <-> ( x e. RR /\ A <_ x /\ x < B ) ) )
2	1	biimp3a	|- ( ( A e. RR /\ B e. RR* /\ x e. ( A [,) B ) ) -> ( x e. RR /\ A <_ x /\ x < B ) )
3	2	simp1d	|- ( ( A e. RR /\ B e. RR* /\ x e. ( A [,) B ) ) -> x e. RR )
4	3	3expia	|- ( ( A e. RR /\ B e. RR* ) -> ( x e. ( A [,) B ) -> x e. RR ) )
5	4	ssrdv	|- ( ( A e. RR /\ B e. RR* ) -> ( A [,) B ) C_ RR )