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Theorem renemnf 7281
Description: No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
Assertion
Ref Expression
renemnf  |-  ( A  e.  RR  ->  A  =/= -oo )

Proof of Theorem renemnf
StepHypRef Expression
1 mnfnre 7275 . . . 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   -oocmnf 7265
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-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-setind 4308  ax-cnex 7181  ax-resscn 7182
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  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-ral 2358  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-uni 3622  df-pnf 7269  df-mnf 7270
This theorem is referenced by:  renemnfd  7284  renfdisj  7291  ltxrlt  7297  xrnemnf  8981  xrlttri3  9000  ngtmnft  9013  xrrebnd  9014  rexneg  9025
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