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Theorem ixxf 11302
Description: The set of intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)
Hypothesis
Ref Expression
ixx.1  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
Assertion
Ref Expression
ixxf  |-  O :
( RR*  X.  RR* ) --> ~P RR*
Distinct variable groups:    x, y,
z, R    x, S, y, z
Allowed substitution hints:    O( x, y, z)

Proof of Theorem ixxf
StepHypRef Expression
1 ssrab2 3432 . . . 4  |-  { z  e.  RR*  |  (
x R z  /\  z S y ) } 
C_  RR*
2 xrex 10980 . . . . 5  |-  RR*  e.  _V
32elpw2 4451 . . . 4  |-  ( { z  e.  RR*  |  ( x R z  /\  z S y ) }  e.  ~P RR*  <->  { z  e.  RR*  |  ( x R z  /\  z S y ) } 
C_  RR* )
41, 3mpbir 209 . . 3  |-  { z  e.  RR*  |  (
x R z  /\  z S y ) }  e.  ~P RR*
54rgen2w 2779 . 2  |-  A. x  e.  RR*  A. y  e. 
RR*  { z  e.  RR*  |  ( x R z  /\  z S y ) }  e.  ~P RR*
6 ixx.1 . . 3  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
76fmpt2 6636 . 2  |-  ( A. x  e.  RR*  A. y  e.  RR*  { z  e. 
RR*  |  ( x R z  /\  z S y ) }  e.  ~P RR*  <->  O :
( RR*  X.  RR* ) --> ~P RR* )
85, 7mpbi 208 1  |-  O :
( RR*  X.  RR* ) --> ~P RR*
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2710   {crab 2714    C_ wss 3323   ~Pcpw 3855   class class class wbr 4287    X. cxp 4833   -->wf 5409    e. cmpt2 6088   RR*cxr 9409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-xr 9414
This theorem is referenced by:  ixxex  11303  ixxssxr  11304  elixx3g  11305  ndmioo  11319  iccf  11380  iocpnfordt  18799  icomnfordt  18800  tpr2rico  26311
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