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

Theorem renepnf 8879
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 8874 . . . 4  |-  +oo  e/  RR
2 df-nel 2449 . . . 4  |-  (  +oo  e/  RR  <->  -.  +oo  e.  RR )
31, 2mpbi 199 . . 3  |-  -.  +oo  e.  RR
4 eleq1 2343 . . 3  |-  ( A  =  +oo  ->  ( A  e.  RR  <->  +oo  e.  RR ) )
53, 4mtbiri 294 . 2  |-  ( A  =  +oo  ->  -.  A  e.  RR )
65necon2ai 2491 1  |-  ( A  e.  RR  ->  A  =/=  +oo )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    e. wcel 1684    =/= wne 2446    e/ wnel 2447   RRcr 8736    +oocpnf 8864
This theorem is referenced by:  renepnfd  8882  renfdisj  8885  xrnepnf  10461  rexneg  10538  rexadd  10559  xaddnepnf  10562  xaddcom  10565  xaddid1  10566  xnegdi  10568  xpncan  10571  xleadd1a  10573  rexmul  10591  xmulpnf1  10594  xadddilem  10614  rpsup  10970
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-pr 3647  df-uni 3828  df-pnf 8869
  Copyright terms: Public domain W3C validator