MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iccsupr Structured version   Unicode version

Theorem iccsupr 10999
Description: A nonempty subset of a closed real interval satisfies the conditions for the existence of its supremum (see suprcl 9970). (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 10994 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
2 sstr 3358 . . . . 5  |-  ( ( S  C_  ( A [,] B )  /\  ( A [,] B )  C_  RR )  ->  S  C_  RR )
32ancoms 441 . . . 4  |-  ( ( ( A [,] B
)  C_  RR  /\  S  C_  ( A [,] B
) )  ->  S  C_  RR )
41, 3sylan 459 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  S  C_  RR )
543adant3 978 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B )  /\  C  e.  S )  ->  S  C_  RR )
6 ne0i 3636 . . 3  |-  ( C  e.  S  ->  S  =/=  (/) )
763ad2ant3 981 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B )  /\  C  e.  S )  ->  S  =/=  (/) )
8 simplr 733 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  B  e.  RR )
9 ssel 3344 . . . . . . . 8  |-  ( S 
C_  ( A [,] B )  ->  (
y  e.  S  -> 
y  e.  ( A [,] B ) ) )
10 elicc2 10977 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( y  e.  ( A [,] B )  <-> 
( y  e.  RR  /\  A  <_  y  /\  y  <_  B ) ) )
1110biimpd 200 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( y  e.  ( A [,] B )  ->  ( y  e.  RR  /\  A  <_ 
y  /\  y  <_  B ) ) )
129, 11sylan9r 641 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  ( y  e.  S  ->  ( y  e.  RR  /\  A  <_ 
y  /\  y  <_  B ) ) )
1312imp 420 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B
) )  /\  y  e.  S )  ->  (
y  e.  RR  /\  A  <_  y  /\  y  <_  B ) )
1413simp3d 972 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B
) )  /\  y  e.  S )  ->  y  <_  B )
1514ralrimiva 2791 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  A. y  e.  S  y  <_  B )
16 breq2 4218 . . . . . 6  |-  ( x  =  B  ->  (
y  <_  x  <->  y  <_  B ) )
1716ralbidv 2727 . . . . 5  |-  ( x  =  B  ->  ( A. y  e.  S  y  <_  x  <->  A. y  e.  S  y  <_  B ) )
1817rspcev 3054 . . . 4  |-  ( ( B  e.  RR  /\  A. y  e.  S  y  <_  B )  ->  E. x  e.  RR  A. y  e.  S  y  <_  x )
198, 15, 18syl2anc 644 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  E. x  e.  RR  A. y  e.  S  y  <_  x )
20193adant3 978 . 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 1135 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708    C_ wss 3322   (/)c0 3630   class class class wbr 4214  (class class class)co 6083   RRcr 8991    <_ cle 9123   [,]cicc 10921
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-pre-lttri 9066  ax-pre-lttrn 9067
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-icc 10925
  Copyright terms: Public domain W3C validator