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Theorem renepnf 8004
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 7998 . . . 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 7809   +oocpnf 7988
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 4121  ax-un 4433  ax-cnex 7901  ax-resscn 7902
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 2739  df-in 3135  df-ss 3142  df-pw 3577  df-uni 3810  df-pnf 7993
This theorem is referenced by:  renepnfd  8007  renfdisj  8016  ltxrlt  8022  xrnepnf  9777  xrlttri3  9796  nltpnft  9813  xrrebnd  9818  rexneg  9829  xrpnfdc  9841  rexadd  9851  xaddnepnf  9857  xaddcom  9860  xaddid1  9861  xnn0xadd0  9866  xnegdi  9867  xpncan  9870  xleadd1a  9872  xltadd1  9875  xsubge0  9880  xposdif  9881  xleaddadd  9886  xrmaxrecl  11262
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