Metamath Proof Explorer


Theorem iocgtlb

Description: An element of a left-open right-closed interval is larger than its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019)

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

Proof

Step Hyp Ref Expression
1 elioc1
 |-  ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A (,] B ) <-> ( C e. RR* /\ A < C /\ C <_ B ) ) )
2 simp2
 |-  ( ( C e. RR* /\ A < C /\ C <_ B ) -> A < C )
3 1 2 syl6bi
 |-  ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A (,] B ) -> A < C ) )
4 3 3impia
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. ( A (,] B ) ) -> A < C )