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Theorem ixxf 10819
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 3344 . . . 4  |-  { z  e.  RR*  |  (
x R z  /\  z S y ) } 
C_  RR*
2 xrex 10502 . . . . 5  |-  RR*  e.  _V
32elpw2 4277 . . . 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 200 . . 3  |-  { z  e.  RR*  |  (
x R z  /\  z S y ) }  e.  ~P RR*
54rgen2w 2696 . 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 6318 . 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 199 1  |-  O :
( RR*  X.  RR* ) --> ~P RR*
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1647    e. wcel 1715   A.wral 2628   {crab 2632    C_ wss 3238   ~Pcpw 3714   class class class wbr 4125    X. cxp 4790   -->wf 5354    e. cmpt2 5983   RR*cxr 9013
This theorem is referenced by:  ixxex  10820  ixxssxr  10821  elixx3g  10822  ndmioo  10836  iccf  10895  iocpnfordt  17162  icomnfordt  17163  tpr2rico  23786
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fv 5366  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-xr 9018
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