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Theorem renepnf 7073
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 7067 . . . 4  |- +oo  e/  RR
21neli 2299 . . 3  |-  -. +oo  e.  RR
3 eleq1 2100 . . 3  |-  ( A  = +oo  ->  ( A  e.  RR  <-> +oo  e.  RR ) )
42, 3mtbiri 600 . 2  |-  ( A  = +oo  ->  -.  A  e.  RR )
54necon2ai 2259 1  |-  ( A  e.  RR  ->  A  =/= +oo )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    e. wcel 1393    =/= wne 2204   RRcr 6888   +oocpnf 7057
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-un 4170  ax-cnex 6975  ax-resscn 6976
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-rex 2312  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-uni 3581  df-pnf 7062
This theorem is referenced by:  renepnfd  7076  renfdisj  7079  ltxrlt  7085  xrnepnf  8700  xrlttri3  8718  nltpnft  8730  xrrebnd  8732  rexneg  8743
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