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Theorem renemnf 7536
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 7530 . . . 4  |- -oo  e/  RR
21neli 2352 . . 3  |-  -. -oo  e.  RR
3 eleq1 2150 . . 3  |-  ( A  = -oo  ->  ( A  e.  RR  <-> -oo  e.  RR ) )
42, 3mtbiri 635 . 2  |-  ( A  = -oo  ->  -.  A  e.  RR )
54necon2ai 2309 1  |-  ( A  e.  RR  ->  A  =/= -oo )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438    =/= wne 2255   RRcr 7349   -oocmnf 7520
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-setind 4353  ax-cnex 7436  ax-resscn 7437
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-uni 3654  df-pnf 7524  df-mnf 7525
This theorem is referenced by:  renemnfd  7539  renfdisj  7546  ltxrlt  7552  xrnemnf  9248  xrlttri3  9267  ngtmnft  9280  xrrebnd  9281  rexneg  9292
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