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Theorem suprfinzcl 10975
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 9654 . . . . . 6  |-  <  Or  RR
3 soss 4807 . . . . . 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 996 . . . 4  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  A  e.  Fin )
7 simp2 995 . . . 4  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  A  =/=  (/) )
8 simp1 994 . . . 4  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  A  C_  ZZ )
9 fisup2g 7918 . . . 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 1228 . . 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 3501 . . . . . 6  |-  ( A 
C_  ZZ  ->  A  C_  RR )
13123ad2ant1 1015 . . . . 5  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  A  C_  RR )
14 ssrexv 3551 . . . . 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 3484 . . . . . . . . . . . . . . 15  |-  ( ( A  C_  ZZ  /\  a  e.  A )  ->  a  e.  ZZ )
1716zred 10965 . . . . . . . . . . . . . 14  |-  ( ( A  C_  ZZ  /\  a  e.  A )  ->  a  e.  RR )
1817ex 432 . . . . . . . . . . . . 13  |-  ( A 
C_  ZZ  ->  ( a  e.  A  ->  a  e.  RR ) )
19183ad2ant1 1015 . . . . . . . . . . . 12  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  (
a  e.  A  -> 
a  e.  RR ) )
2019adantr 463 . . . . . . . . . . 11  |-  ( ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e.  Fin )  /\  r  e.  RR )  ->  (
a  e.  A  -> 
a  e.  RR ) )
2120imp 427 . . . . . . . . . 10  |-  ( ( ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e.  Fin )  /\  r  e.  RR )  /\  a  e.  A )  ->  a  e.  RR )
22 simplr 753 . . . . . . . . . 10  |-  ( ( ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e.  Fin )  /\  r  e.  RR )  /\  a  e.  A )  ->  r  e.  RR )
2321, 22lenltd 9720 . . . . . . . . 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 2890 . . . . . . 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 466 . . . . 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 2929 . . . 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 10938 . 2  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  E. r  e.  RR  A. a  e.  A  a  <_  r
)  ->  sup ( A ,  RR ,  <  )  e.  A )
3230, 31syld3an3 1271 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 367    /\ w3a 971    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805    C_ wss 3461   (/)c0 3783   class class class wbr 4439    Or wor 4788   Fincfn 7509   supcsup 7892   RRcr 9480    < clt 9617    <_ cle 9618   ZZcz 10860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-1o 7122  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861
This theorem is referenced by: (None)
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