Metamath Proof Explorer


Theorem eliccxrd

Description: Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses eliccxrd.1
|- ( ph -> A e. RR* )
eliccxrd.2
|- ( ph -> B e. RR* )
eliccxrd.3
|- ( ph -> C e. RR* )
eliccxrd.4
|- ( ph -> A <_ C )
eliccxrd.5
|- ( ph -> C <_ B )
Assertion eliccxrd
|- ( ph -> C e. ( A [,] B ) )

Proof

Step Hyp Ref Expression
1 eliccxrd.1
 |-  ( ph -> A e. RR* )
2 eliccxrd.2
 |-  ( ph -> B e. RR* )
3 eliccxrd.3
 |-  ( ph -> C e. RR* )
4 eliccxrd.4
 |-  ( ph -> A <_ C )
5 eliccxrd.5
 |-  ( ph -> C <_ B )
6 4 5 jca
 |-  ( ph -> ( A <_ C /\ C <_ B ) )
7 elicc4
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C e. ( A [,] B ) <-> ( A <_ C /\ C <_ B ) ) )
8 1 2 3 7 syl3anc
 |-  ( ph -> ( C e. ( A [,] B ) <-> ( A <_ C /\ C <_ B ) ) )
9 6 8 mpbird
 |-  ( ph -> C e. ( A [,] B ) )