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Theorem suprubex 8867
Description: A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by Jim Kingdon, 18-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 )
suprubex.b  |-  ( ph  ->  B  e.  A )
Assertion
Ref Expression
suprubex  |-  ( ph  ->  B  <_  sup ( A ,  RR ,  <  ) )
Distinct variable groups:    x, A, y, z    ph, x
Allowed substitution hints:    ph( y, z)    B( x, y, z)

Proof of Theorem suprubex
Dummy variables  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 suprubex.ss . . 3  |-  ( ph  ->  A  C_  RR )
2 suprubex.b . . 3  |-  ( ph  ->  B  e.  A )
31, 2sseldd 3148 . 2  |-  ( ph  ->  B  e.  RR )
4 lttri3 7999 . . . 4  |-  ( ( f  e.  RR  /\  g  e.  RR )  ->  ( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
54adantl 275 . . 3  |-  ( (
ph  /\  ( f  e.  RR  /\  g  e.  RR ) )  -> 
( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
6 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 ) ) )
75, 6supclti 6975 . 2  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  RR )
85, 6supubti 6976 . . 3  |-  ( ph  ->  ( B  e.  A  ->  -.  sup ( A ,  RR ,  <  )  <  B ) )
92, 8mpd 13 . 2  |-  ( ph  ->  -.  sup ( A ,  RR ,  <  )  <  B )
103, 7, 9nltled 8040 1  |-  ( ph  ->  B  <_  sup ( A ,  RR ,  <  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 2141   A.wral 2448   E.wrex 2449    C_ wss 3121   class class class wbr 3989   supcsup 6959   RRcr 7773    < clt 7954    <_ cle 7955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-pre-ltirr 7886  ax-pre-apti 7889
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-iota 5160  df-riota 5809  df-sup 6961  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960
This theorem is referenced by:  suprzclex  9310
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