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Theorem suprlubex 8167
Description: The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by Jim Kingdon, 19-Jan-2022.)
Hypotheses
Ref Expression
suprubex.ex  |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. z  e.  A  y  <  z ) ) )
suprubex.ss  |-  ( ph  ->  A  C_  RR )
suprlubex.b  |-  ( ph  ->  B  e.  RR )
Assertion
Ref Expression
suprlubex  |-  ( ph  ->  ( B  <  sup ( A ,  RR ,  <  )  <->  E. z  e.  A  B  <  z ) )
Distinct variable groups:    x, A, y, z    ph, x    z, B
Allowed substitution hints:    ph( y, z)    B( x, y)

Proof of Theorem suprlubex
Dummy variables  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 suprlubex.b . 2  |-  ( ph  ->  B  e.  RR )
2 lttri3 7328 . . . 4  |-  ( ( f  e.  RR  /\  g  e.  RR )  ->  ( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
32adantl 271 . . 3  |-  ( (
ph  /\  ( f  e.  RR  /\  g  e.  RR ) )  -> 
( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
4 suprubex.ex . . 3  |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. z  e.  A  y  <  z ) ) )
5 ltso 7326 . . . 4  |-  <  Or  RR
65a1i 9 . . 3  |-  ( ph  ->  <  Or  RR )
7 suprubex.ss . . 3  |-  ( ph  ->  A  C_  RR )
83, 4, 6, 7suplub2ti 6509 . 2  |-  ( (
ph  /\  B  e.  RR )  ->  ( B  <  sup ( A ,  RR ,  <  )  <->  E. z  e.  A  B  <  z ) )
91, 8mpdan 412 1  |-  ( ph  ->  ( B  <  sup ( A ,  RR ,  <  )  <->  E. z  e.  A  B  <  z ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1434   A.wral 2353   E.wrex 2354    C_ wss 2982   class class class wbr 3805    Or wor 4078   supcsup 6490   RRcr 7112    < clt 7285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7199  ax-resscn 7200  ax-pre-ltirr 7220  ax-pre-ltwlin 7221  ax-pre-lttrn 7222  ax-pre-apti 7223
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-po 4079  df-iso 4080  df-xp 4397  df-iota 4917  df-riota 5520  df-sup 6492  df-pnf 7287  df-mnf 7288  df-ltxr 7290
This theorem is referenced by:  suprnubex  8168  suprzclex  8596
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