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Metamath Proof Explorer

Theorem supicclub2

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 )
supicclub2.1 
|- ( ( ph /\ z e. A ) -> z <_ D )
Assertion supicclub2 
|- ( ph -> sup ( A , RR , < ) <_ D )

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 supicclub2.1 
 |-  ( ( ph /\ z e. A ) -> z <_ D )
7 iccssxr 
 |-  ( B [,] C ) C_ RR*
8 1 2 3 4 supicc 
 |-  ( ph -> sup ( A , RR , < ) e. ( B [,] C ) )
9 7 8 sselid 
 |-  ( ph -> sup ( A , RR , < ) e. RR* )
10 3 7 sstrdi 
 |-  ( ph -> A C_ RR* )
11 10 5 sseldd 
 |-  ( ph -> D e. RR* )
12 10 sselda 
 |-  ( ( ph /\ z e. A ) -> z e. RR* )
13 11 adantr 
 |-  ( ( ph /\ z e. A ) -> D e. RR* )
14 12 13 xrlenltd 
 |-  ( ( ph /\ z e. A ) -> ( z <_ D <-> -. D < z ) )
15 6 14 mpbid 
 |-  ( ( ph /\ z e. A ) -> -. D < z )
16 15 nrexdv 
 |-  ( ph -> -. E. z e. A D < z )
17 1 2 3 4 5 supicclub 
 |-  ( ph -> ( D < sup ( A , RR , < ) <-> E. z e. A D < z ) )
18 16 17 mtbird 
 |-  ( ph -> -. D < sup ( A , RR , < ) )
19 9 11 18 xrnltled 
 |-  ( ph -> sup ( A , RR , < ) <_ D )