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

Theorem renepnf 7280
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 7274 . . . 4  |- +oo  e/  RR
21neli 2346 . . 3  |-  -. +oo  e.  RR
3 eleq1 2145 . . 3  |-  ( A  = +oo  ->  ( A  e.  RR  <-> +oo  e.  RR ) )
42, 3mtbiri 633 . 2  |-  ( A  = +oo  ->  -.  A  e.  RR )
54necon2ai 2303 1  |-  ( A  e.  RR  ->  A  =/= +oo )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434    =/= wne 2249   RRcr 7094   +oocpnf 7264
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-un 4216  ax-cnex 7181  ax-resscn 7182
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-rex 2359  df-rab 2362  df-v 2612  df-in 2988  df-ss 2995  df-pw 3402  df-uni 3622  df-pnf 7269
This theorem is referenced by:  renepnfd  7283  renfdisj  7291  ltxrlt  7297  xrnepnf  8982  xrlttri3  9000  nltpnft  9012  xrrebnd  9014  rexneg  9025
  Copyright terms: Public domain W3C validator