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Theorem ixxex 9286
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 9274 . . . 4  |-  RR*  e.  _V
21, 1xpex 4541 . . 3  |-  ( RR*  X. 
RR* )  e.  _V
31pwex 4009 . . 3  |-  ~P RR*  e.  _V
42, 3xpex 4541 . 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 9285 . . 3  |-  O :
( RR*  X.  RR* ) --> ~P RR*
7 fssxp 5163 . . 3  |-  ( O : ( RR*  X.  RR* )
--> ~P RR*  ->  O  C_  ( ( RR*  X.  RR* )  X.  ~P RR* )
)
86, 7ax-mp 7 . 2  |-  O  C_  ( ( RR*  X.  RR* )  X.  ~P RR* )
94, 8ssexi 3969 1  |-  O  e. 
_V
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1289    e. wcel 1438   {crab 2363   _Vcvv 2619    C_ wss 2997   ~Pcpw 3425   class class class wbr 3837    X. cxp 4426   -->wf 4998    |-> cmpt2 5636   RR*cxr 7500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-cnex 7415  ax-resscn 7416
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-pnf 7503  df-mnf 7504  df-xr 7505
This theorem is referenced by:  iooex  9294
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