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Theorem ixxex 9873
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 9830 . . . 4  |-  RR*  e.  _V
21, 1xpex 4737 . . 3  |-  ( RR*  X. 
RR* )  e.  _V
31pwex 4180 . . 3  |-  ~P RR*  e.  _V
42, 3xpex 4737 . 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 9872 . . 3  |-  O :
( RR*  X.  RR* ) --> ~P RR*
7 fssxp 5378 . . 3  |-  ( O : ( RR*  X.  RR* )
--> ~P RR*  ->  O  C_  ( ( RR*  X.  RR* )  X.  ~P RR* )
)
86, 7ax-mp 5 . 2  |-  O  C_  ( ( RR*  X.  RR* )  X.  ~P RR* )
94, 8ssexi 4138 1  |-  O  e. 
_V
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1353    e. wcel 2148   {crab 2459   _Vcvv 2737    C_ wss 3129   ~Pcpw 3574   class class class wbr 4000    X. cxp 4620   -->wf 5207    e. cmpo 5870   RR*cxr 7968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-pnf 7971  df-mnf 7972  df-xr 7973
This theorem is referenced by:  iooex  9881
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