MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  renfdisj Structured version   Unicode version

Theorem renfdisj 9138
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 3668 . 2  |-  ( ( RR  i^i  {  +oo , 
-oo } )  =  (/)  <->  A. x  e.  RR  -.  x  e.  {  +oo ,  -oo } )
2 vex 2959 . . . . 5  |-  x  e. 
_V
32elpr 3832 . . . 4  |-  ( x  e.  {  +oo ,  -oo }  <->  ( x  = 
+oo  \/  x  =  -oo ) )
4 renepnf 9132 . . . . . 6  |-  ( x  e.  RR  ->  x  =/=  +oo )
54necon2bi 2650 . . . . 5  |-  ( x  =  +oo  ->  -.  x  e.  RR )
6 renemnf 9133 . . . . . 6  |-  ( x  e.  RR  ->  x  =/=  -oo )
76necon2bi 2650 . . . . 5  |-  ( x  =  -oo  ->  -.  x  e.  RR )
85, 7jaoi 369 . . . 4  |-  ( ( x  =  +oo  \/  x  =  -oo )  ->  -.  x  e.  RR )
93, 8sylbi 188 . . 3  |-  ( x  e.  {  +oo ,  -oo }  ->  -.  x  e.  RR )
109con2i 114 . 2  |-  ( x  e.  RR  ->  -.  x  e.  {  +oo ,  -oo } )
111, 10mprgbir 2776 1  |-  ( RR 
i^i  {  +oo ,  -oo } )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 358    = wceq 1652    e. wcel 1725    i^i cin 3319   (/)c0 3628   {cpr 3815   RRcr 8989    +oocpnf 9117    -oocmnf 9118
This theorem is referenced by:  ssxr  9145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123
  Copyright terms: Public domain W3C validator