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Theorem renepnf 8895
Description: No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
Assertion
Ref Expression
renepnf  |-  ( A  e.  RR  ->  A  =/=  +oo )

Proof of Theorem renepnf
StepHypRef Expression
1 pnfnre 8890 . . . 4  |-  +oo  e/  RR
2 df-nel 2462 . . . 4  |-  (  +oo  e/  RR  <->  -.  +oo  e.  RR )
31, 2mpbi 199 . . 3  |-  -.  +oo  e.  RR
4 eleq1 2356 . . 3  |-  ( A  =  +oo  ->  ( A  e.  RR  <->  +oo  e.  RR ) )
53, 4mtbiri 294 . 2  |-  ( A  =  +oo  ->  -.  A  e.  RR )
65necon2ai 2504 1  |-  ( A  e.  RR  ->  A  =/=  +oo )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    e. wcel 1696    =/= wne 2459    e/ wnel 2460   RRcr 8752    +oocpnf 8880
This theorem is referenced by:  renepnfd  8898  renfdisj  8901  xrnepnf  10477  rexneg  10554  rexadd  10575  xaddnepnf  10578  xaddcom  10581  xaddid1  10582  xnegdi  10584  xpncan  10587  xleadd1a  10589  rexmul  10607  xmulpnf1  10610  xadddilem  10630  rpsup  10986  xaddeq0  23319  ovoliunnfl  25001
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-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-pw 3640  df-sn 3659  df-pr 3660  df-uni 3844  df-pnf 8885
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