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Theorem ixxssxr 9898
Description: The set of intervals of extended reals maps to subsets of extended reals. (Contributed by Mario Carneiro, 4-Jul-2014.)
Hypothesis
Ref Expression
ixxssxr.1  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
Assertion
Ref Expression
ixxssxr  |-  ( A O B )  C_  RR*
Distinct variable groups:    x, y, z, R    x, S, y, z    x, A, y, z    x, B, y, z    x, O, y, z

Proof of Theorem ixxssxr
StepHypRef Expression
1 ixxssxr.1 . . . 4  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
21elmpocl 6068 . . 3  |-  ( x  e.  ( A O B )  ->  ( A  e.  RR*  /\  B  e.  RR* ) )
31ixxf 9896 . . . . . 6  |-  O :
( RR*  X.  RR* ) --> ~P RR*
43fovcl 5979 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A O B )  e. 
~P RR* )
54elpwid 3586 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A O B )  C_  RR* )
65sseld 3154 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A O B )  ->  x  e.  RR* ) )
72, 6mpcom 36 . 2  |-  ( x  e.  ( A O B )  ->  x  e.  RR* )
87ssriv 3159 1  |-  ( A O B )  C_  RR*
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1353    e. wcel 2148   {crab 2459    C_ wss 3129   ~Pcpw 3575   class class class wbr 4003  (class class class)co 5874    e. cmpo 5876   RR*cxr 7989
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 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-cnex 7901  ax-resscn 7902
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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-pnf 7992  df-mnf 7993  df-xr 7994
This theorem is referenced by:  iccssxr  9954  iocssxr  9955  icossxr  9956
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