ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ixxex Unicode version

Theorem ixxex 9650
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 9607 . . . 4  |-  RR*  e.  _V
21, 1xpex 4624 . . 3  |-  ( RR*  X. 
RR* )  e.  _V
31pwex 4077 . . 3  |-  ~P RR*  e.  _V
42, 3xpex 4624 . 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 9649 . . 3  |-  O :
( RR*  X.  RR* ) --> ~P RR*
7 fssxp 5260 . . 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 4036 1  |-  O  e. 
_V
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1316    e. wcel 1465   {crab 2397   _Vcvv 2660    C_ wss 3041   ~Pcpw 3480   class class class wbr 3899    X. cxp 4507   -->wf 5089    e. cmpo 5744   RR*cxr 7767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-cnex 7679  ax-resscn 7680
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-pnf 7770  df-mnf 7771  df-xr 7772
This theorem is referenced by:  iooex  9658
  Copyright terms: Public domain W3C validator