This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem gtnelicc

Description: A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses gtnelicc.a 
|- ( ph -> A e. RR* )
gtnelicc.b 
|- ( ph -> B e. RR )
gtnelicc.c 
|- ( ph -> C e. RR* )
gtnelicc.bltc 
|- ( ph -> B < C )
Assertion gtnelicc 
|- ( ph -> -. C e. ( A [,] B ) )

Proof

Step Hyp Ref Expression
1 gtnelicc.a 
 |-  ( ph -> A e. RR* )
2 gtnelicc.b 
 |-  ( ph -> B e. RR )
3 gtnelicc.c 
 |-  ( ph -> C e. RR* )
4 gtnelicc.bltc 
 |-  ( ph -> B < C )
5 2 rexrd 
 |-  ( ph -> B e. RR* )
6 xrltnle 
 |-  ( ( B e. RR* /\ C e. RR* ) -> ( B < C <-> -. C <_ B ) )
7 5 3 6 syl2anc 
 |-  ( ph -> ( B < C <-> -. C <_ B ) )
8 4 7 mpbid 
 |-  ( ph -> -. C <_ B )
9 8 intnand 
 |-  ( ph -> -. ( A <_ C /\ C <_ B ) )
10 elicc4 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C e. ( A [,] B ) <-> ( A <_ C /\ C <_ B ) ) )
11 1 5 3 10 syl3anc 
 |-  ( ph -> ( C e. ( A [,] B ) <-> ( A <_ C /\ C <_ B ) ) )
12 9 11 mtbird 
 |-  ( ph -> -. C e. ( A [,] B ) )