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Metamath Proof Explorer

Theorem ltnelicc

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

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

Proof

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