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Theorem renfdisj 7972
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 3462 . 2  |-  ( ( RR  i^i  { +oo , -oo } )  =  (/) 
<-> 
A. x  e.  RR  -.  x  e.  { +oo , -oo } )
2 vex 2733 . . . . 5  |-  x  e. 
_V
32elpr 3602 . . . 4  |-  ( x  e.  { +oo , -oo }  <->  ( x  = +oo  \/  x  = -oo ) )
4 renepnf 7960 . . . . . 6  |-  ( x  e.  RR  ->  x  =/= +oo )
54necon2bi 2395 . . . . 5  |-  ( x  = +oo  ->  -.  x  e.  RR )
6 renemnf 7961 . . . . . 6  |-  ( x  e.  RR  ->  x  =/= -oo )
76necon2bi 2395 . . . . 5  |-  ( x  = -oo  ->  -.  x  e.  RR )
85, 7jaoi 711 . . . 4  |-  ( ( x  = +oo  \/  x  = -oo )  ->  -.  x  e.  RR )
93, 8sylbi 120 . . 3  |-  ( x  e.  { +oo , -oo }  ->  -.  x  e.  RR )
109con2i 622 . 2  |-  ( x  e.  RR  ->  -.  x  e.  { +oo , -oo } )
111, 10mprgbir 2528 1  |-  ( RR 
i^i  { +oo , -oo } )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 703    = wceq 1348    e. wcel 2141    i^i cin 3120   (/)c0 3414   {cpr 3582   RRcr 7766   +oocpnf 7944   -oocmnf 7945
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-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-un 4416  ax-setind 4519  ax-cnex 7858  ax-resscn 7859
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-uni 3795  df-pnf 7949  df-mnf 7950
This theorem is referenced by: (None)
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