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Theorem iccsupr 11629
Description: A nonempty subset of a closed real interval satisfies the conditions for the existence of its supremum (see suprcl 10515). (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 11618 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
2 sstr 3517 . . . . 5  |-  ( ( S  C_  ( A [,] B )  /\  ( A [,] B )  C_  RR )  ->  S  C_  RR )
32ancoms 453 . . . 4  |-  ( ( ( A [,] B
)  C_  RR  /\  S  C_  ( A [,] B
) )  ->  S  C_  RR )
41, 3sylan 471 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  S  C_  RR )
543adant3 1016 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B )  /\  C  e.  S )  ->  S  C_  RR )
6 ne0i 3796 . . 3  |-  ( C  e.  S  ->  S  =/=  (/) )
763ad2ant3 1019 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B )  /\  C  e.  S )  ->  S  =/=  (/) )
8 simplr 754 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  B  e.  RR )
9 ssel 3503 . . . . . . . 8  |-  ( S 
C_  ( A [,] B )  ->  (
y  e.  S  -> 
y  e.  ( A [,] B ) ) )
10 elicc2 11601 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( y  e.  ( A [,] B )  <-> 
( y  e.  RR  /\  A  <_  y  /\  y  <_  B ) ) )
1110biimpd 207 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( y  e.  ( A [,] B )  ->  ( y  e.  RR  /\  A  <_ 
y  /\  y  <_  B ) ) )
129, 11sylan9r 658 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  ( y  e.  S  ->  ( y  e.  RR  /\  A  <_ 
y  /\  y  <_  B ) ) )
1312imp 429 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B
) )  /\  y  e.  S )  ->  (
y  e.  RR  /\  A  <_  y  /\  y  <_  B ) )
1413simp3d 1010 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B
) )  /\  y  e.  S )  ->  y  <_  B )
1514ralrimiva 2881 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  A. y  e.  S  y  <_  B )
16 breq2 4457 . . . . . 6  |-  ( x  =  B  ->  (
y  <_  x  <->  y  <_  B ) )
1716ralbidv 2906 . . . . 5  |-  ( x  =  B  ->  ( A. y  e.  S  y  <_  x  <->  A. y  e.  S  y  <_  B ) )
1817rspcev 3219 . . . 4  |-  ( ( B  e.  RR  /\  A. y  e.  S  y  <_  B )  ->  E. x  e.  RR  A. y  e.  S  y  <_  x )
198, 15, 18syl2anc 661 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  E. x  e.  RR  A. y  e.  S  y  <_  x )
20193adant3 1016 . 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 1176 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818    C_ wss 3481   (/)c0 3790   class class class wbr 4453  (class class class)co 6295   RRcr 9503    <_ cle 9641   [,]cicc 11544
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-pre-lttri 9578  ax-pre-lttrn 9579
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-icc 11548
This theorem is referenced by:  supicc  11680
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