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Theorem renepnf 9132
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 9127 . . . 4  |-  +oo  e/  RR
21neli 2697 . . 3  |-  -.  +oo  e.  RR
3 eleq1 2496 . . 3  |-  ( A  =  +oo  ->  ( A  e.  RR  <->  +oo  e.  RR ) )
42, 3mtbiri 295 . 2  |-  ( A  =  +oo  ->  -.  A  e.  RR )
54necon2ai 2649 1  |-  ( A  e.  RR  ->  A  =/=  +oo )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    =/= wne 2599   RRcr 8989    +oocpnf 9117
This theorem is referenced by:  renepnfd  9135  renfdisj  9138  xrnepnf  10719  rexneg  10797  rexadd  10818  xaddnepnf  10821  xaddcom  10824  xaddid1  10825  xnegdi  10827  xpncan  10830  xleadd1a  10832  rexmul  10850  xmulpnf1  10853  xadddilem  10873  rpsup  11247  hash1snb  11681  xaddeq0  24119  ovoliunnfl  26248  voliunnfl  26250  volsupnfl  26251  euhash1  28167
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-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-pw 3801  df-sn 3820  df-pr 3821  df-uni 4016  df-pnf 9122
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