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Theorem renfdisj 9659
Description: The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
Assertion
Ref Expression
renfdisj  |-  ( RR 
i^i  { +oo , -oo } )  =  (/)

Proof of Theorem renfdisj
StepHypRef Expression
1 disj 3872 . 2  |-  ( ( RR  i^i  { +oo , -oo } )  =  (/) 
<-> 
A. x  e.  RR  -.  x  e.  { +oo , -oo } )
2 vex 3121 . . . . 5  |-  x  e. 
_V
32elpr 4051 . . . 4  |-  ( x  e.  { +oo , -oo }  <->  ( x  = +oo  \/  x  = -oo ) )
4 renepnf 9653 . . . . . 6  |-  ( x  e.  RR  ->  x  =/= +oo )
54necon2bi 2704 . . . . 5  |-  ( x  = +oo  ->  -.  x  e.  RR )
6 renemnf 9654 . . . . . 6  |-  ( x  e.  RR  ->  x  =/= -oo )
76necon2bi 2704 . . . . 5  |-  ( x  = -oo  ->  -.  x  e.  RR )
85, 7jaoi 379 . . . 4  |-  ( ( x  = +oo  \/  x  = -oo )  ->  -.  x  e.  RR )
93, 8sylbi 195 . . 3  |-  ( x  e.  { +oo , -oo }  ->  -.  x  e.  RR )
109con2i 120 . 2  |-  ( x  e.  RR  ->  -.  x  e.  { +oo , -oo } )
111, 10mprgbir 2831 1  |-  ( RR 
i^i  { +oo , -oo } )  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 368    = wceq 1379    e. wcel 1767    i^i cin 3480   (/)c0 3790   {cpr 4035   RRcr 9503   +oocpnf 9637   -oocmnf 9638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-resscn 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643
This theorem is referenced by:  ssxr  9666
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