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

Theorem iccsupr 10736
Description: A nonempty subset of a closed real interval satisfies the conditions for the existence of its supremum (see suprcl 9714). (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 10731 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
2 sstr 3187 . . . . 5  |-  ( ( S  C_  ( A [,] B )  /\  ( A [,] B )  C_  RR )  ->  S  C_  RR )
32ancoms 439 . . . 4  |-  ( ( ( A [,] B
)  C_  RR  /\  S  C_  ( A [,] B
) )  ->  S  C_  RR )
41, 3sylan 457 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  S  C_  RR )
543adant3 975 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B )  /\  C  e.  S )  ->  S  C_  RR )
6 ne0i 3461 . . 3  |-  ( C  e.  S  ->  S  =/=  (/) )
763ad2ant3 978 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B )  /\  C  e.  S )  ->  S  =/=  (/) )
8 simplr 731 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  B  e.  RR )
9 ssel 3174 . . . . . . . 8  |-  ( S 
C_  ( A [,] B )  ->  (
y  e.  S  -> 
y  e.  ( A [,] B ) ) )
10 elicc2 10715 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( y  e.  ( A [,] B )  <-> 
( y  e.  RR  /\  A  <_  y  /\  y  <_  B ) ) )
1110biimpd 198 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( y  e.  ( A [,] B )  ->  ( y  e.  RR  /\  A  <_ 
y  /\  y  <_  B ) ) )
129, 11sylan9r 639 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  ( y  e.  S  ->  ( y  e.  RR  /\  A  <_ 
y  /\  y  <_  B ) ) )
1312imp 418 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B
) )  /\  y  e.  S )  ->  (
y  e.  RR  /\  A  <_  y  /\  y  <_  B ) )
1413simp3d 969 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B
) )  /\  y  e.  S )  ->  y  <_  B )
1514ralrimiva 2626 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  A. y  e.  S  y  <_  B )
16 breq2 4027 . . . . . 6  |-  ( x  =  B  ->  (
y  <_  x  <->  y  <_  B ) )
1716ralbidv 2563 . . . . 5  |-  ( x  =  B  ->  ( A. y  e.  S  y  <_  x  <->  A. y  e.  S  y  <_  B ) )
1817rspcev 2884 . . . 4  |-  ( ( B  e.  RR  /\  A. y  e.  S  y  <_  B )  ->  E. x  e.  RR  A. y  e.  S  y  <_  x )
198, 15, 18syl2anc 642 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  S  C_  ( A [,] B ) )  ->  E. x  e.  RR  A. y  e.  S  y  <_  x )
20193adant3 975 . 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 1132 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    C_ wss 3152   (/)c0 3455   class class class wbr 4023  (class class class)co 5858   RRcr 8736    <_ cle 8868   [,]cicc 10659
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-icc 10663
  Copyright terms: Public domain W3C validator