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Theorem ixxf 8997
Description: The set of intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)
Hypothesis
Ref Expression
ixx.1  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
Assertion
Ref Expression
ixxf  |-  O :
( RR*  X.  RR* ) --> ~P RR*
Distinct variable groups:    x, y, z, R    x, S, y, z
Allowed substitution hints:    O( x, y, z)

Proof of Theorem ixxf
StepHypRef Expression
1 ssrab2 3080 . . . 4  |-  { z  e.  RR*  |  (
x R z  /\  z S y ) } 
C_  RR*
2 xrex 8986 . . . . 5  |-  RR*  e.  _V
32elpw2 3940 . . . 4  |-  ( { z  e.  RR*  |  ( x R z  /\  z S y ) }  e.  ~P RR*  <->  { z  e.  RR*  |  ( x R z  /\  z S y ) } 
C_  RR* )
41, 3mpbir 144 . . 3  |-  { z  e.  RR*  |  (
x R z  /\  z S y ) }  e.  ~P RR*
54rgen2w 2420 . 2  |-  A. x  e.  RR*  A. y  e. 
RR*  { z  e.  RR*  |  ( x R z  /\  z S y ) }  e.  ~P RR*
6 ixx.1 . . 3  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
76fmpt2 5858 . 2  |-  ( A. x  e.  RR*  A. y  e.  RR*  { z  e. 
RR*  |  ( x R z  /\  z S y ) }  e.  ~P RR*  <->  O :
( RR*  X.  RR* ) --> ~P RR* )
85, 7mpbi 143 1  |-  O :
( RR*  X.  RR* ) --> ~P RR*
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285    e. wcel 1434   A.wral 2349   {crab 2353    C_ wss 2974   ~Pcpw 3390   class class class wbr 3793    X. cxp 4369   -->wf 4928    |-> cmpt2 5545   RR*cxr 7214
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-cnex 7129  ax-resscn 7130
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fv 4940  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-pnf 7217  df-mnf 7218  df-xr 7219
This theorem is referenced by:  ixxex  8998  ixxssxr  8999  iccf  9071
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