Theorem iccssre

Description: A closed real interval is a set of reals. (Contributed by FL, 6-Jun-2007) (Proof shortened by Paul Chapman, 21-Jan-2008)

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

Proof

Step	Hyp	Ref	Expression
1		elicc2	|- ( ( 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 )