Theorem iocssre

Description: A closed-above interval with real upper bound is a set of reals. (Contributed by FL, 29-May-2014)

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

Proof

Step	Hyp	Ref	Expression
1		elioc2	|- ( ( 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 )