Theorem iccgelb

Description: An element of a closed interval is more than or equal to its lower bound. (Contributed by Thierry Arnoux, 23-Dec-2016)

	Ref	Expression
Assertion	iccgelb	|- ( ( A e. RR* /\ B e. RR* /\ C e. ( A [,] B ) ) -> A <_ C )

Proof

Step	Hyp	Ref	Expression
1		elicc1	|- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A [,] B ) <-> ( C e. RR* /\ A <_ C /\ C <_ B ) ) )
2	1	biimpa	|- ( ( ( A e. RR* /\ B e. RR* ) /\ C e. ( A [,] B ) ) -> ( C e. RR* /\ A <_ C /\ C <_ B ) )
3	2	simp2d	|- ( ( ( A e. RR* /\ B e. RR* ) /\ C e. ( A [,] B ) ) -> A <_ C )
4	3	3impa	|- ( ( A e. RR* /\ B e. RR* /\ C e. ( A [,] B ) ) -> A <_ C )