ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  renepnf Unicode version

Theorem renepnf 7514
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 7508 . . . 4  |- +oo  e/  RR
21neli 2352 . . 3  |-  -. +oo  e.  RR
3 eleq1 2150 . . 3  |-  ( A  = +oo  ->  ( A  e.  RR  <-> +oo  e.  RR ) )
42, 3mtbiri 635 . 2  |-  ( A  = +oo  ->  -.  A  e.  RR )
54necon2ai 2309 1  |-  ( A  e.  RR  ->  A  =/= +oo )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438    =/= wne 2255   RRcr 7328   +oocpnf 7498
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-un 4251  ax-cnex 7415  ax-resscn 7416
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-rex 2365  df-rab 2368  df-v 2621  df-in 3003  df-ss 3010  df-pw 3427  df-uni 3649  df-pnf 7503
This theorem is referenced by:  renepnfd  7517  renfdisj  7525  ltxrlt  7531  xrnepnf  9218  xrlttri3  9236  nltpnft  9248  xrrebnd  9250  rexneg  9261
  Copyright terms: Public domain W3C validator