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Theorem iccsupr 8605
Description: A nonempty subset of a closed real interval satisfies the conditions for the existence of its supremum. To be useful without excluded middle, we'll probably need to change not equal to apart, and perhaps make other changes, but the theorem does hold as stated here. (Contributed by Paul Chapman, 21-Jan-2008.)
Assertion
Ref Expression
iccsupr  RR  RR  S  C_  [,]  C  S  S  C_  RR  S  =/=  (/)  RR  S  <_
Distinct variable groups:   ,   ,,   , S,
Allowed substitution hints:   ()    C(,)

Proof of Theorem iccsupr
StepHypRef Expression
1 iccssre 8594 . . . 4  RR  RR  [,]  C_  RR
2 sstr 2947 . . . . 5  S  C_  [,]  [,]  C_  RR  S  C_  RR
32ancoms 255 . . . 4  [,]  C_  RR  S  C_  [,]  S  C_  RR
41, 3sylan 267 . . 3  RR  RR  S  C_  [,]  S  C_  RR
543adant3 923 . 2  RR  RR  S  C_  [,]  C  S  S  C_  RR
6 ne0i 3224 . . 3  C  S  S  =/=  (/)
763ad2ant3 926 . 2  RR  RR  S  C_  [,]  C  S  S  =/=  (/)
8 simplr 482 . . . 4  RR  RR  S  C_  [,]  RR
9 ssel 2933 . . . . . . . 8  S 
C_  [,]  S  [,]
10 elicc2 8577 . . . . . . . . 9  RR  RR  [,]  RR  <_  <_
1110biimpd 132 . . . . . . . 8  RR  RR  [,]  RR  <_  <_
129, 11sylan9r 390 . . . . . . 7  RR  RR  S  C_  [,]  S  RR  <_  <_
1312imp 115 . . . . . 6  RR  RR  S  C_  [,]  S  RR  <_  <_
1413simp3d 917 . . . . 5  RR  RR  S  C_  [,]  S  <_
1514ralrimiva 2386 . . . 4  RR  RR  S  C_  [,]  S  <_
16 breq2 3759 . . . . . 6  <_  <_
1716ralbidv 2320 . . . . 5  S  <_  S  <_
1817rspcev 2650 . . . 4  RR  S 
<_  RR  S  <_
198, 15, 18syl2anc 391 . . 3  RR  RR  S  C_  [,]  RR  S 
<_
20193adant3 923 . 2  RR  RR  S  C_  [,]  C  S  RR  S 
<_
215, 7, 203jca 1083 1  RR  RR  S  C_  [,]  C  S  S  C_  RR  S  =/=  (/)  RR  S  <_
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   w3a 884   wceq 1242   wcel 1390    =/= wne 2201  wral 2300  wrex 2301    C_ wss 2911   (/)c0 3218   class class class wbr 3755  (class class class)co 5455   RRcr 6710    <_ cle 6858   [,]cicc 8530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6774  ax-resscn 6775  ax-pre-ltirr 6795  ax-pre-ltwlin 6796  ax-pre-lttrn 6797
This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-po 4024  df-iso 4025  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862  df-le 6863  df-icc 8534
This theorem is referenced by: (None)
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