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Metamath Proof Explorer

Theorem supicclub

Description: The supremum of a bounded set of real numbers is the least upper bound. (Contributed by Thierry Arnoux, 23-May-2019)

Ref Expression
Hypotheses supicc.1 
|- ( ph -> B e. RR )
supicc.2 
|- ( ph -> C e. RR )
supicc.3 
|- ( ph -> A C_ ( B [,] C ) )
supicc.4 
|- ( ph -> A =/= (/) )
supiccub.1 
|- ( ph -> D e. A )
Assertion supicclub 
|- ( ph -> ( D < sup ( A , RR , < ) <-> E. z e. A D < z ) )

Proof

Step Hyp Ref Expression
1 supicc.1 
 |-  ( ph -> B e. RR )
2 supicc.2 
 |-  ( ph -> C e. RR )
3 supicc.3 
 |-  ( ph -> A C_ ( B [,] C ) )
4 supicc.4 
 |-  ( ph -> A =/= (/) )
5 supiccub.1 
 |-  ( ph -> D e. A )
6 iccssre 
 |-  ( ( B e. RR /\ C e. RR ) -> ( B [,] C ) C_ RR )
7 1 2 6 syl2anc 
 |-  ( ph -> ( B [,] C ) C_ RR )
8 3 7 sstrd 
 |-  ( ph -> A C_ RR )
9 1 adantr 
 |-  ( ( ph /\ x e. A ) -> B e. RR )
10 9 rexrd 
 |-  ( ( ph /\ x e. A ) -> B e. RR* )
11 2 adantr 
 |-  ( ( ph /\ x e. A ) -> C e. RR )
12 11 rexrd 
 |-  ( ( ph /\ x e. A ) -> C e. RR* )
13 3 sselda 
 |-  ( ( ph /\ x e. A ) -> x e. ( B [,] C ) )
14 iccleub 
 |-  ( ( B e. RR* /\ C e. RR* /\ x e. ( B [,] C ) ) -> x <_ C )
15 10 12 13 14 syl3anc 
 |-  ( ( ph /\ x e. A ) -> x <_ C )
16 15 ralrimiva 
 |-  ( ph -> A. x e. A x <_ C )
17 brralrspcev 
 |-  ( ( C e. RR /\ A. x e. A x <_ C ) -> E. y e. RR A. x e. A x <_ y )
18 2 16 17 syl2anc 
 |-  ( ph -> E. y e. RR A. x e. A x <_ y )
19 8 5 sseldd 
 |-  ( ph -> D e. RR )
20 suprlub 
 |-  ( ( ( A C_ RR /\ A =/= (/) /\ E. y e. RR A. x e. A x <_ y ) /\ D e. RR ) -> ( D < sup ( A , RR , < ) <-> E. z e. A D < z ) )
21 8 4 18 19 20 syl31anc 
 |-  ( ph -> ( D < sup ( A , RR , < ) <-> E. z e. A D < z ) )