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Theorem suprubex 8324
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 3013 . 2  |-  ( ph  ->  B  e.  RR )
4 lttri3 7486 . . . 4  |-  ( ( f  e.  RR  /\  g  e.  RR )  ->  ( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
54adantl 271 . . 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 6614 . 2  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  RR )
85, 6supubti 6615 . . 3  |-  ( ph  ->  ( B  e.  A  ->  -.  sup ( A ,  RR ,  <  )  <  B ) )
92, 8mpd 13 . 2  |-  ( ph  ->  -.  sup ( A ,  RR ,  <  )  <  B )
103, 7, 9nltled 7525 1  |-  ( ph  ->  B  <_  sup ( A ,  RR ,  <  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1436   A.wral 2355   E.wrex 2356    C_ wss 2986   class class class wbr 3814   supcsup 6598   RRcr 7270    < clt 7443    <_ cle 7444
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-cnex 7357  ax-resscn 7358  ax-pre-ltirr 7378  ax-pre-apti 7381
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2616  df-sbc 2829  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-br 3815  df-opab 3869  df-xp 4410  df-cnv 4412  df-iota 4937  df-riota 5550  df-sup 6600  df-pnf 7445  df-mnf 7446  df-xr 7447  df-ltxr 7448  df-le 7449
This theorem is referenced by:  suprzclex  8754
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