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Metamath Proof Explorer

Theorem icoub

Description: A left-closed, right-open interval does not contain its upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Assertion icoub 
|- ( A e. RR* -> -. B e. ( A [,) B ) )

Proof

Step Hyp Ref Expression
1 simpl 
 |-  ( ( A e. RR* /\ B e. ( A [,) B ) ) -> A e. RR* )
2 icossxr 
 |-  ( A [,) B ) C_ RR*
3 id 
 |-  ( B e. ( A [,) B ) -> B e. ( A [,) B ) )
4 2 3 sselid 
 |-  ( B e. ( A [,) B ) -> B e. RR* )
5 4 adantl 
 |-  ( ( A e. RR* /\ B e. ( A [,) B ) ) -> B e. RR* )
6 simpr 
 |-  ( ( A e. RR* /\ B e. ( A [,) B ) ) -> B e. ( A [,) B ) )
7 icoltub 
 |-  ( ( A e. RR* /\ B e. RR* /\ B e. ( A [,) B ) ) -> B < B )
8 1 5 6 7 syl3anc 
 |-  ( ( A e. RR* /\ B e. ( A [,) B ) ) -> B < B )
9 xrltnr 
 |-  ( B e. RR* -> -. B < B )
10 4 9 syl 
 |-  ( B e. ( A [,) B ) -> -. B < B )
11 10 adantl 
 |-  ( ( A e. RR* /\ B e. ( A [,) B ) ) -> -. B < B )
12 8 11 pm2.65da 
 |-  ( A e. RR* -> -. B e. ( A [,) B ) )