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Theorem renepnf 9630
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 9624 . . . 4  |- +oo  e/  RR
21neli 2795 . . 3  |-  -. +oo  e.  RR
3 eleq1 2532 . . 3  |-  ( A  = +oo  ->  ( A  e.  RR  <-> +oo  e.  RR ) )
42, 3mtbiri 303 . 2  |-  ( A  = +oo  ->  -.  A  e.  RR )
54necon2ai 2695 1  |-  ( A  e.  RR  ->  A  =/= +oo )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762    =/= wne 2655   RRcr 9480   +oocpnf 9614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-resscn 9538
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-pw 4005  df-sn 4021  df-pr 4023  df-uni 4239  df-pnf 9619
This theorem is referenced by:  renepnfd  9633  renfdisj  9636  xrnepnf  11318  rexneg  11399  rexadd  11420  xaddnepnf  11423  xaddcom  11426  xaddid1  11427  xnegdi  11429  xpncan  11432  xleadd1a  11434  rexmul  11452  xmulpnf1  11455  xadddilem  11475  rpsup  11949  hashneq0  12389  hash1snb  12431  euhash1  12432  xaddeq0  27227  ovoliunnfl  29620  voliunnfl  29622  volsupnfl  29623
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