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Theorem ixxex 10919
Description: The set of intervals of extended reals exists. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
Hypothesis
Ref Expression
ixx.1  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
Assertion
Ref Expression
ixxex  |-  O  e. 
_V
Distinct variable groups:    x, y,
z, R    x, S, y, z
Allowed substitution hints:    O( x, y, z)

Proof of Theorem ixxex
StepHypRef Expression
1 xrex 10601 . . . 4  |-  RR*  e.  _V
21, 1xpex 4982 . . 3  |-  ( RR*  X. 
RR* )  e.  _V
31pwex 4374 . . 3  |-  ~P RR*  e.  _V
42, 3xpex 4982 . 2  |-  ( (
RR*  X.  RR* )  X. 
~P RR* )  e.  _V
5 ixx.1 . . . 4  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
65ixxf 10918 . . 3  |-  O :
( RR*  X.  RR* ) --> ~P RR*
7 fssxp 5594 . . 3  |-  ( O : ( RR*  X.  RR* )
--> ~P RR*  ->  O  C_  ( ( RR*  X.  RR* )  X.  ~P RR* )
)
86, 7ax-mp 8 . 2  |-  O  C_  ( ( RR*  X.  RR* )  X.  ~P RR* )
94, 8ssexi 4340 1  |-  O  e. 
_V
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791   class class class wbr 4204    X. cxp 4868   -->wf 5442    e. cmpt2 6075   RR*cxr 9111
This theorem is referenced by:  iooex  10931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-xr 9116
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