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Theorem suprfinzcl 10971
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 10867 . . . . . 6  |-  ZZ  C_  RR
2 ltso 9661 . . . . . 6  |-  <  Or  RR
3 soss 4818 . . . . . 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 998 . . . 4  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  A  e.  Fin )
7 simp2 997 . . . 4  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  A  =/=  (/) )
8 simp1 996 . . . 4  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  A  C_  ZZ )
9 fisup2g 7922 . . . 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 1230 . . 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 3516 . . . . . 6  |-  ( A 
C_  ZZ  ->  A  C_  RR )
13123ad2ant1 1017 . . . . 5  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  A  C_  RR )
14 ssrexv 3565 . . . . 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 3499 . . . . . . . . . . . . . . 15  |-  ( ( A  C_  ZZ  /\  a  e.  A )  ->  a  e.  ZZ )
1716zred 10962 . . . . . . . . . . . . . 14  |-  ( ( A  C_  ZZ  /\  a  e.  A )  ->  a  e.  RR )
1817ex 434 . . . . . . . . . . . . 13  |-  ( A 
C_  ZZ  ->  ( a  e.  A  ->  a  e.  RR ) )
19183ad2ant1 1017 . . . . . . . . . . . 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 9726 . . . . . . . . 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 2900 . . . . . . 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 2938 . . . 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 10936 . 2  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  E. r  e.  RR  A. a  e.  A  a  <_  r
)  ->  sup ( A ,  RR ,  <  )  e.  A )
3230, 31syld3an3 1273 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 973    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815    C_ wss 3476   (/)c0 3785   class class class wbr 4447    Or wor 4799   Fincfn 7513   supcsup 7896   RRcr 9487    < clt 9624    <_ cle 9625   ZZcz 10860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-1o 7127  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861
This theorem is referenced by: (None)
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