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Theorem renfdisj 7239
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 3299 . 2  |-  ( ( RR  i^i  { +oo , -oo } )  =  (/) 
<-> 
A. x  e.  RR  -.  x  e.  { +oo , -oo } )
2 vex 2605 . . . . 5  |-  x  e. 
_V
32elpr 3427 . . . 4  |-  ( x  e.  { +oo , -oo }  <->  ( x  = +oo  \/  x  = -oo ) )
4 renepnf 7228 . . . . . 6  |-  ( x  e.  RR  ->  x  =/= +oo )
54necon2bi 2301 . . . . 5  |-  ( x  = +oo  ->  -.  x  e.  RR )
6 renemnf 7229 . . . . . 6  |-  ( x  e.  RR  ->  x  =/= -oo )
76necon2bi 2301 . . . . 5  |-  ( x  = -oo  ->  -.  x  e.  RR )
85, 7jaoi 669 . . . 4  |-  ( ( x  = +oo  \/  x  = -oo )  ->  -.  x  e.  RR )
93, 8sylbi 119 . . 3  |-  ( x  e.  { +oo , -oo }  ->  -.  x  e.  RR )
109con2i 590 . 2  |-  ( x  e.  RR  ->  -.  x  e.  { +oo , -oo } )
111, 10mprgbir 2422 1  |-  ( RR 
i^i  { +oo , -oo } )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 662    = wceq 1285    e. wcel 1434    i^i cin 2973   (/)c0 3258   {cpr 3407   RRcr 7042   +oocpnf 7212   -oocmnf 7213
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-in1 577  ax-in2 578  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-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-uni 3610  df-pnf 7217  df-mnf 7218
This theorem is referenced by: (None)
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