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Theorem ixxf 10882
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 3388 . . . 4  |-  { z  e.  RR*  |  (
x R z  /\  z S y ) } 
C_  RR*
2 xrex 10565 . . . . 5  |-  RR*  e.  _V
32elpw2 4324 . . . 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 201 . . 3  |-  { z  e.  RR*  |  (
x R z  /\  z S y ) }  e.  ~P RR*
54rgen2w 2734 . 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 6377 . 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 200 1  |-  O :
( RR*  X.  RR* ) --> ~P RR*
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   {crab 2670    C_ wss 3280   ~Pcpw 3759   class class class wbr 4172    X. cxp 4835   -->wf 5409    e. cmpt2 6042   RR*cxr 9075
This theorem is referenced by:  ixxex  10883  ixxssxr  10884  elixx3g  10885  ndmioo  10899  iccf  10959  iocpnfordt  17233  icomnfordt  17234  tpr2rico  24263
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-xr 9080
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