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Theorem iccsupr 8936
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  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B )  /\  C  e.  S )  ->  ( S  C_  RR  /\  S  =/=  (/)  /\  E. x  e.  RR  A. y  e.  S  y  <_  x ) )
Distinct variable groups:    y, A    x, B, y    x, S, y
Allowed substitution hints:    A( x)    C( x, y)

Proof of Theorem iccsupr
StepHypRef Expression
1 iccssre 8925 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
2 sstr 2981 . . . . 5  |-  ( ( S  C_  ( A [,] B )  /\  ( A [,] B )  C_  RR )  ->  S  C_  RR )
32ancoms 259 . . . 4  |-  ( ( ( A [,] B
)  C_  RR  /\  S  C_  ( A [,] B
) )  ->  S  C_  RR )
41, 3sylan 271 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  S  C_  RR )
543adant3 935 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B )  /\  C  e.  S )  ->  S  C_  RR )
6 ne0i 3258 . . 3  |-  ( C  e.  S  ->  S  =/=  (/) )
763ad2ant3 938 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B )  /\  C  e.  S )  ->  S  =/=  (/) )
8 simplr 490 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  B  e.  RR )
9 ssel 2967 . . . . . . . 8  |-  ( S 
C_  ( A [,] B )  ->  (
y  e.  S  -> 
y  e.  ( A [,] B ) ) )
10 elicc2 8908 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( y  e.  ( A [,] B )  <-> 
( y  e.  RR  /\  A  <_  y  /\  y  <_  B ) ) )
1110biimpd 136 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( y  e.  ( A [,] B )  ->  ( y  e.  RR  /\  A  <_ 
y  /\  y  <_  B ) ) )
129, 11sylan9r 396 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  ( y  e.  S  ->  ( y  e.  RR  /\  A  <_ 
y  /\  y  <_  B ) ) )
1312imp 119 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B
) )  /\  y  e.  S )  ->  (
y  e.  RR  /\  A  <_  y  /\  y  <_  B ) )
1413simp3d 929 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B
) )  /\  y  e.  S )  ->  y  <_  B )
1514ralrimiva 2409 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  A. y  e.  S  y  <_  B )
16 breq2 3796 . . . . . 6  |-  ( x  =  B  ->  (
y  <_  x  <->  y  <_  B ) )
1716ralbidv 2343 . . . . 5  |-  ( x  =  B  ->  ( A. y  e.  S  y  <_  x  <->  A. y  e.  S  y  <_  B ) )
1817rspcev 2673 . . . 4  |-  ( ( B  e.  RR  /\  A. y  e.  S  y  <_  B )  ->  E. x  e.  RR  A. y  e.  S  y  <_  x )
198, 15, 18syl2anc 397 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  E. x  e.  RR  A. y  e.  S  y  <_  x )
20193adant3 935 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B )  /\  C  e.  S )  ->  E. x  e.  RR  A. y  e.  S  y  <_  x )
215, 7, 203jca 1095 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B )  /\  C  e.  S )  ->  ( S  C_  RR  /\  S  =/=  (/)  /\  E. x  e.  RR  A. y  e.  S  y  <_  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    /\ w3a 896    = wceq 1259    e. wcel 1409    =/= wne 2220   A.wral 2323   E.wrex 2324    C_ wss 2945   (/)c0 3252   class class class wbr 3792  (class class class)co 5540   RRcr 6946    <_ cle 7120   [,]cicc 8861
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-po 4061  df-iso 4062  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-icc 8865
This theorem is referenced by: (None)
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