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Theorem ixxf 11542
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 3571 . . . 4  |-  { z  e.  RR*  |  (
x R z  /\  z S y ) } 
C_  RR*
2 xrex 11218 . . . . 5  |-  RR*  e.  _V
32elpw2 4601 . . . 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 2816 . 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 6840 . 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808    C_ wss 3461   ~Pcpw 3999   class class class wbr 4439    X. cxp 4986   -->wf 5566    |-> cmpt2 6272   RR*cxr 9616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-xr 9621
This theorem is referenced by:  ixxex  11543  ixxssxr  11544  elixx3g  11545  ndmioo  11559  iccf  11626  iocpnfordt  19883  icomnfordt  19884  tpr2rico  28129
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