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Theorem iccsupr 10922
Description: A nonempty subset of a closed real interval satisfies the conditions for the existence of its supremum (see suprcl 9893). (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 10917 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
2 sstr 3292 . . . . 5  |-  ( ( S  C_  ( A [,] B )  /\  ( A [,] B )  C_  RR )  ->  S  C_  RR )
32ancoms 440 . . . 4  |-  ( ( ( A [,] B
)  C_  RR  /\  S  C_  ( A [,] B
) )  ->  S  C_  RR )
41, 3sylan 458 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  S  C_  RR )
543adant3 977 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B )  /\  C  e.  S )  ->  S  C_  RR )
6 ne0i 3570 . . 3  |-  ( C  e.  S  ->  S  =/=  (/) )
763ad2ant3 980 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B )  /\  C  e.  S )  ->  S  =/=  (/) )
8 simplr 732 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  B  e.  RR )
9 ssel 3278 . . . . . . . 8  |-  ( S 
C_  ( A [,] B )  ->  (
y  e.  S  -> 
y  e.  ( A [,] B ) ) )
10 elicc2 10900 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( y  e.  ( A [,] B )  <-> 
( y  e.  RR  /\  A  <_  y  /\  y  <_  B ) ) )
1110biimpd 199 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( y  e.  ( A [,] B )  ->  ( y  e.  RR  /\  A  <_ 
y  /\  y  <_  B ) ) )
129, 11sylan9r 640 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  ( y  e.  S  ->  ( y  e.  RR  /\  A  <_ 
y  /\  y  <_  B ) ) )
1312imp 419 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B
) )  /\  y  e.  S )  ->  (
y  e.  RR  /\  A  <_  y  /\  y  <_  B ) )
1413simp3d 971 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B
) )  /\  y  e.  S )  ->  y  <_  B )
1514ralrimiva 2725 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  A. y  e.  S  y  <_  B )
16 breq2 4150 . . . . . 6  |-  ( x  =  B  ->  (
y  <_  x  <->  y  <_  B ) )
1716ralbidv 2662 . . . . 5  |-  ( x  =  B  ->  ( A. y  e.  S  y  <_  x  <->  A. y  e.  S  y  <_  B ) )
1817rspcev 2988 . . . 4  |-  ( ( B  e.  RR  /\  A. y  e.  S  y  <_  B )  ->  E. x  e.  RR  A. y  e.  S  y  <_  x )
198, 15, 18syl2anc 643 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  E. x  e.  RR  A. y  e.  S  y  <_  x )
20193adant3 977 . 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 1134 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 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   A.wral 2642   E.wrex 2643    C_ wss 3256   (/)c0 3564   class class class wbr 4146  (class class class)co 6013   RRcr 8915    <_ cle 9047   [,]cicc 10844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-pre-lttri 8990  ax-pre-lttrn 8991
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-icc 10848
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