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Theorem renepnf 8005
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 7999 . . . 4  |- +oo  e/  RR
21neli 2444 . . 3  |-  -. +oo  e.  RR
3 eleq1 2240 . . 3  |-  ( A  = +oo  ->  ( A  e.  RR  <-> +oo  e.  RR ) )
42, 3mtbiri 675 . 2  |-  ( A  = +oo  ->  -.  A  e.  RR )
54necon2ai 2401 1  |-  ( A  e.  RR  ->  A  =/= +oo )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148    =/= wne 2347   RRcr 7810   +oocpnf 7989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-un 4434  ax-cnex 7902  ax-resscn 7903
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-rex 2461  df-rab 2464  df-v 2740  df-in 3136  df-ss 3143  df-pw 3578  df-uni 3811  df-pnf 7994
This theorem is referenced by:  renepnfd  8008  renfdisj  8017  ltxrlt  8023  xrnepnf  9778  xrlttri3  9797  nltpnft  9814  xrrebnd  9819  rexneg  9830  xrpnfdc  9842  rexadd  9852  xaddnepnf  9858  xaddcom  9861  xaddid1  9862  xnn0xadd0  9867  xnegdi  9868  xpncan  9871  xleadd1a  9873  xltadd1  9876  xsubge0  9881  xposdif  9882  xleaddadd  9887  xrmaxrecl  11263
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