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Theorem suprzclex 8943
Description: The supremum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 20-Dec-2021.)
Hypotheses
Ref Expression
suprzclex.ex  |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. z  e.  A  y  <  z ) ) )
suprzclex.ss  |-  ( ph  ->  A  C_  ZZ )
Assertion
Ref Expression
suprzclex  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  A )
Distinct variable groups:    x, A, y, z    ph, x, z
Allowed substitution hint:    ph( y)

Proof of Theorem suprzclex
Dummy variables  w  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lttri3 7662 . . . . . 6  |-  ( ( f  e.  RR  /\  g  e.  RR )  ->  ( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
21adantl 272 . . . . 5  |-  ( (
ph  /\  ( f  e.  RR  /\  g  e.  RR ) )  -> 
( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
3 suprzclex.ex . . . . 5  |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. z  e.  A  y  <  z ) ) )
42, 3supclti 6773 . . . 4  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  RR )
54ltm1d 8490 . . 3  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  -  1 )  <  sup ( A ,  RR ,  <  ) )
6 suprzclex.ss . . . . 5  |-  ( ph  ->  A  C_  ZZ )
7 zssre 8855 . . . . 5  |-  ZZ  C_  RR
86, 7syl6ss 3051 . . . 4  |-  ( ph  ->  A  C_  RR )
9 peano2rem 7846 . . . . 5  |-  ( sup ( A ,  RR ,  <  )  e.  RR  ->  ( sup ( A ,  RR ,  <  )  -  1 )  e.  RR )
104, 9syl 14 . . . 4  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  -  1 )  e.  RR )
113, 8, 10suprlubex 8510 . . 3  |-  ( ph  ->  ( ( sup ( A ,  RR ,  <  )  -  1 )  <  sup ( A ,  RR ,  <  )  <->  E. z  e.  A  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )
125, 11mpbid 146 . 2  |-  ( ph  ->  E. z  e.  A  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
)
136adantr 271 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  A  C_  ZZ )
1413sselda 3039 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  e.  ZZ )
157, 14sseldi 3037 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  e.  RR )
164adantr 271 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  sup ( A ,  RR ,  <  )  e.  RR )
1716adantr 271 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  sup ( A ,  RR ,  <  )  e.  RR )
18 simprl 499 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  z  e.  A )
1913, 18sseldd 3040 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  z  e.  ZZ )
20 zre 8852 . . . . . . . . . . 11  |-  ( z  e.  ZZ  ->  z  e.  RR )
2119, 20syl 14 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  z  e.  RR )
22 peano2re 7715 . . . . . . . . . 10  |-  ( z  e.  RR  ->  (
z  +  1 )  e.  RR )
2321, 22syl 14 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  ( z  +  1 )  e.  RR )
2423adantr 271 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  (
z  +  1 )  e.  RR )
253ad2antrr 473 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  ( y  < 
x  ->  E. z  e.  A  y  <  z ) ) )
268ad2antrr 473 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  A  C_  RR )
27 simpr 109 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  e.  A )
2825, 26, 27suprubex 8509 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  <_  sup ( A ,  RR ,  <  ) )
29 simprr 500 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  ( sup ( A ,  RR ,  <  )  -  1 )  <  z )
30 1red 7600 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  1  e.  RR )
3116, 30, 21ltsubaddd 8115 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  ( ( sup ( A ,  RR ,  <  )  -  1 )  <  z  <->  sup ( A ,  RR ,  <  )  <  ( z  +  1 ) ) )
3229, 31mpbid 146 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  sup ( A ,  RR ,  <  )  <  ( z  +  1 ) )
3332adantr 271 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  sup ( A ,  RR ,  <  )  <  ( z  +  1 ) )
3415, 17, 24, 28, 33lelttrd 7705 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  <  ( z  +  1 ) )
3519adantr 271 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  z  e.  ZZ )
36 zleltp1 8903 . . . . . . . 8  |-  ( ( w  e.  ZZ  /\  z  e.  ZZ )  ->  ( w  <_  z  <->  w  <  ( z  +  1 ) ) )
3714, 35, 36syl2anc 404 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  (
w  <_  z  <->  w  <  ( z  +  1 ) ) )
3834, 37mpbird 166 . . . . . 6  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  <_  z )
3938ralrimiva 2458 . . . . 5  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  A. w  e.  A  w  <_  z )
40 breq2 3871 . . . . . . . . . . . . 13  |-  ( z  =  w  ->  (
y  <  z  <->  y  <  w ) )
4140cbvrexv 2605 . . . . . . . . . . . 12  |-  ( E. z  e.  A  y  <  z  <->  E. w  e.  A  y  <  w )
4241imbi2i 225 . . . . . . . . . . 11  |-  ( ( y  <  x  ->  E. z  e.  A  y  <  z )  <->  ( y  <  x  ->  E. w  e.  A  y  <  w ) )
4342ralbii 2395 . . . . . . . . . 10  |-  ( A. y  e.  RR  (
y  <  x  ->  E. z  e.  A  y  <  z )  <->  A. y  e.  RR  ( y  < 
x  ->  E. w  e.  A  y  <  w ) )
4443anbi2i 446 . . . . . . . . 9  |-  ( ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. z  e.  A  y  <  z ) )  <-> 
( A. y  e.  A  -.  x  < 
y  /\  A. y  e.  RR  ( y  < 
x  ->  E. w  e.  A  y  <  w ) ) )
4544rexbii 2396 . . . . . . . 8  |-  ( E. x  e.  RR  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. z  e.  A  y  <  z ) )  <->  E. x  e.  RR  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. w  e.  A  y  <  w ) ) )
463, 45sylib 121 . . . . . . 7  |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. w  e.  A  y  <  w ) ) )
4746adantr 271 . . . . . 6  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  ( y  < 
x  ->  E. w  e.  A  y  <  w ) ) )
4813, 7syl6ss 3051 . . . . . 6  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  A  C_  RR )
4947, 48, 21suprleubex 8512 . . . . 5  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  ( sup ( A ,  RR ,  <  )  <_  z  <->  A. w  e.  A  w  <_  z ) )
5039, 49mpbird 166 . . . 4  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  sup ( A ,  RR ,  <  )  <_  z )
5147, 48, 18suprubex 8509 . . . 4  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  z  <_  sup ( A ,  RR ,  <  ) )
5216, 21letri3d 7697 . . . 4  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  ( sup ( A ,  RR ,  <  )  =  z  <->  ( sup ( A ,  RR ,  <  )  <_  z  /\  z  <_  sup ( A ,  RR ,  <  ) ) ) )
5350, 51, 52mpbir2and 893 . . 3  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  sup ( A ,  RR ,  <  )  =  z )
5453, 18eqeltrd 2171 . 2  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  sup ( A ,  RR ,  <  )  e.  A )
5512, 54rexlimddv 2507 1  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1296    e. wcel 1445   A.wral 2370   E.wrex 2371    C_ wss 3013   class class class wbr 3867  (class class class)co 5690   supcsup 6757   RRcr 7446   1c1 7448    + caddc 7450    < clt 7619    <_ cle 7620    - cmin 7750   ZZcz 8848
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-addcom 7542  ax-addass 7544  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-0id 7550  ax-rnegex 7551  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558
This theorem depends on definitions:  df-bi 116  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-br 3868  df-opab 3922  df-id 4144  df-po 4147  df-iso 4148  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-sup 6759  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-inn 8521  df-n0 8772  df-z 8849
This theorem is referenced by:  infssuzcldc  11389  gcddvds  11397
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