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Theorem renfdisj 8901
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 3508 . 2  |-  ( ( RR  i^i  {  +oo , 
-oo } )  =  (/)  <->  A. x  e.  RR  -.  x  e.  {  +oo ,  -oo } )
2 vex 2804 . . . . 5  |-  x  e. 
_V
32elpr 3671 . . . 4  |-  ( x  e.  {  +oo ,  -oo }  <->  ( x  = 
+oo  \/  x  =  -oo ) )
4 renepnf 8895 . . . . . 6  |-  ( x  e.  RR  ->  x  =/=  +oo )
54necon2bi 2505 . . . . 5  |-  ( x  =  +oo  ->  -.  x  e.  RR )
6 renemnf 8896 . . . . . 6  |-  ( x  e.  RR  ->  x  =/=  -oo )
76necon2bi 2505 . . . . 5  |-  ( x  =  -oo  ->  -.  x  e.  RR )
85, 7jaoi 368 . . . 4  |-  ( ( x  =  +oo  \/  x  =  -oo )  ->  -.  x  e.  RR )
93, 8sylbi 187 . . 3  |-  ( x  e.  {  +oo ,  -oo }  ->  -.  x  e.  RR )
109con2i 112 . 2  |-  ( x  e.  RR  ->  -.  x  e.  {  +oo ,  -oo } )
111, 10mprgbir 2626 1  |-  ( RR 
i^i  {  +oo ,  -oo } )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 357    = wceq 1632    e. wcel 1696    i^i cin 3164   (/)c0 3468   {cpr 3654   RRcr 8752    +oocpnf 8880    -oocmnf 8881
This theorem is referenced by:  ssxr  8908
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886
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