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Theorem suprfinzcl 30869
Description: The supremum of a nonempty finite set of integers is a member of the set. (Contributed by AV, 1-Oct-2019.)
Assertion
Ref Expression
suprfinzcl  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  sup ( A ,  RR ,  <  )  e.  A )

Proof of Theorem suprfinzcl
Dummy variables  a 
b  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zssre 10740 . . . . . 6  |-  ZZ  C_  RR
2 ltso 9542 . . . . . 6  |-  <  Or  RR
3 soss 4743 . . . . . 6  |-  ( ZZ  C_  RR  ->  (  <  Or  RR  ->  <  Or  ZZ ) )
41, 2, 3mp2 9 . . . . 5  |-  <  Or  ZZ
54a1i 11 . . . 4  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  <  Or  ZZ )
6 simp3 990 . . . 4  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  A  e.  Fin )
7 simp2 989 . . . 4  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  A  =/=  (/) )
8 simp1 988 . . . 4  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  A  C_  ZZ )
9 fisup2g 7803 . . . 4  |-  ( (  <  Or  ZZ  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A  C_  ZZ ) )  ->  E. r  e.  A  ( A. a  e.  A  -.  r  <  a  /\  A. a  e.  ZZ  (
a  <  r  ->  E. b  e.  A  a  <  b ) ) )
105, 6, 7, 8, 9syl13anc 1221 . . 3  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  E. r  e.  A  ( A. a  e.  A  -.  r  <  a  /\  A. a  e.  ZZ  (
a  <  r  ->  E. b  e.  A  a  <  b ) ) )
11 id 22 . . . . . . 7  |-  ( A 
C_  ZZ  ->  A  C_  ZZ )
1211, 1syl6ss 3452 . . . . . 6  |-  ( A 
C_  ZZ  ->  A  C_  RR )
13123ad2ant1 1009 . . . . 5  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  A  C_  RR )
14 ssrexv 3501 . . . . 5  |-  ( A 
C_  RR  ->  ( E. r  e.  A  ( A. a  e.  A  -.  r  <  a  /\  A. a  e.  ZZ  (
a  <  r  ->  E. b  e.  A  a  <  b ) )  ->  E. r  e.  RR  ( A. a  e.  A  -.  r  <  a  /\  A. a  e.  ZZ  (
a  <  r  ->  E. b  e.  A  a  <  b ) ) ) )
1513, 14syl 16 . . . 4  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  ( E. r  e.  A  ( A. a  e.  A  -.  r  <  a  /\  A. a  e.  ZZ  (
a  <  r  ->  E. b  e.  A  a  <  b ) )  ->  E. r  e.  RR  ( A. a  e.  A  -.  r  <  a  /\  A. a  e.  ZZ  (
a  <  r  ->  E. b  e.  A  a  <  b ) ) ) )
16 ssel2 3435 . . . . . . . . . . . . . . 15  |-  ( ( A  C_  ZZ  /\  a  e.  A )  ->  a  e.  ZZ )
1716zred 10834 . . . . . . . . . . . . . 14  |-  ( ( A  C_  ZZ  /\  a  e.  A )  ->  a  e.  RR )
1817ex 434 . . . . . . . . . . . . 13  |-  ( A 
C_  ZZ  ->  ( a  e.  A  ->  a  e.  RR ) )
19183ad2ant1 1009 . . . . . . . . . . . 12  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  (
a  e.  A  -> 
a  e.  RR ) )
2019adantr 465 . . . . . . . . . . 11  |-  ( ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e.  Fin )  /\  r  e.  RR )  ->  (
a  e.  A  -> 
a  e.  RR ) )
2120imp 429 . . . . . . . . . 10  |-  ( ( ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e.  Fin )  /\  r  e.  RR )  /\  a  e.  A )  ->  a  e.  RR )
22 simplr 754 . . . . . . . . . 10  |-  ( ( ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e.  Fin )  /\  r  e.  RR )  /\  a  e.  A )  ->  r  e.  RR )
2321, 22lenltd 9607 . . . . . . . . 9  |-  ( ( ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e.  Fin )  /\  r  e.  RR )  /\  a  e.  A )  ->  (
a  <_  r  <->  -.  r  <  a ) )
2423bicomd 201 . . . . . . . 8  |-  ( ( ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e.  Fin )  /\  r  e.  RR )  /\  a  e.  A )  ->  ( -.  r  <  a  <->  a  <_  r ) )
2524ralbidva 2811 . . . . . . 7  |-  ( ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e.  Fin )  /\  r  e.  RR )  ->  ( A. a  e.  A  -.  r  <  a  <->  A. a  e.  A  a  <_  r ) )
2625biimpd 207 . . . . . 6  |-  ( ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e.  Fin )  /\  r  e.  RR )  ->  ( A. a  e.  A  -.  r  <  a  ->  A. a  e.  A  a  <_  r ) )
2726adantrd 468 . . . . 5  |-  ( ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e.  Fin )  /\  r  e.  RR )  ->  (
( A. a  e.  A  -.  r  < 
a  /\  A. a  e.  ZZ  ( a  < 
r  ->  E. b  e.  A  a  <  b ) )  ->  A. a  e.  A  a  <_  r ) )
2827reximdva 2910 . . . 4  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  ( E. r  e.  RR  ( A. a  e.  A  -.  r  <  a  /\  A. a  e.  ZZ  (
a  <  r  ->  E. b  e.  A  a  <  b ) )  ->  E. r  e.  RR  A. a  e.  A  a  <_  r ) )
2915, 28syld 44 . . 3  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  ( E. r  e.  A  ( A. a  e.  A  -.  r  <  a  /\  A. a  e.  ZZ  (
a  <  r  ->  E. b  e.  A  a  <  b ) )  ->  E. r  e.  RR  A. a  e.  A  a  <_  r ) )
3010, 29mpd 15 . 2  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  E. r  e.  RR  A. a  e.  A  a  <_  r
)
31 suprzcl 10808 . 2  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  E. r  e.  RR  A. a  e.  A  a  <_  r
)  ->  sup ( A ,  RR ,  <  )  e.  A )
3230, 31syld3an3 1264 1  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  sup ( A ,  RR ,  <  )  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    e. wcel 1757    =/= wne 2641   A.wral 2792   E.wrex 2793    C_ wss 3412   (/)c0 3721   class class class wbr 4376    Or wor 4724   Fincfn 7396   supcsup 7777   RRcr 9368    < clt 9505    <_ cle 9506   ZZcz 10733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446  ax-pre-sup 9447
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-recs 6918  df-rdg 6952  df-1o 7006  df-er 7187  df-en 7397  df-dom 7398  df-sdom 7399  df-fin 7400  df-sup 7778  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-nn 10410  df-n0 10667  df-z 10734
This theorem is referenced by: (None)
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