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Theorem suprubex 8677
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 3068 . 2  |-  ( ph  ->  B  e.  RR )
4 lttri3 7812 . . . 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 6853 . 2  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  RR )
85, 6supubti 6854 . . 3  |-  ( ph  ->  ( B  e.  A  ->  -.  sup ( A ,  RR ,  <  )  <  B ) )
92, 8mpd 13 . 2  |-  ( ph  ->  -.  sup ( A ,  RR ,  <  )  <  B )
103, 7, 9nltled 7851 1  |-  ( ph  ->  B  <_  sup ( A ,  RR ,  <  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 1465   A.wral 2393   E.wrex 2394    C_ wss 3041   class class class wbr 3899   supcsup 6837   RRcr 7587    < clt 7768    <_ cle 7769
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-pre-ltirr 7700  ax-pre-apti 7703
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-iota 5058  df-riota 5698  df-sup 6839  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774
This theorem is referenced by:  suprzclex  9117
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