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

Theorem icoval 11450
Description: Value of the closed-below, open-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
Assertion
Ref Expression
icoval  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A [,) B )  =  { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) } )
Distinct variable groups:    x, A    x, B

Proof of Theorem icoval
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ico 11418 . 2  |-  [,)  =  ( y  e.  RR* ,  z  e.  RR*  |->  { x  e.  RR*  |  ( y  <_  x  /\  x  <  z ) } )
21ixxval 11420 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A [,) B )  =  { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   {crab 2803   class class class wbr 4401  (class class class)co 6201   RR*cxr 9529    < clt 9530    <_ cle 9531   [,)cico 11414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-iota 5490  df-fun 5529  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-xr 9534  df-ico 11418
This theorem is referenced by:  ico0  11458  orvcgteel  26995
  Copyright terms: Public domain W3C validator