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Theorem mnfnre 7509
Description: Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
mnfnre  |- -oo  e/  RR

Proof of Theorem mnfnre
StepHypRef Expression
1 cnex 7445 . . . . 5  |-  CC  e.  _V
2 2pwuninelg 6030 . . . . 5  |-  ( CC  e.  _V  ->  -.  ~P ~P U. CC  e.  CC )
31, 2ax-mp 7 . . . 4  |-  -.  ~P ~P U. CC  e.  CC
4 df-mnf 7504 . . . . . 6  |- -oo  =  ~P +oo
5 df-pnf 7503 . . . . . . 7  |- +oo  =  ~P U. CC
65pweqi 3429 . . . . . 6  |-  ~P +oo  =  ~P ~P U. CC
74, 6eqtri 2108 . . . . 5  |- -oo  =  ~P ~P U. CC
87eleq1i 2153 . . . 4  |-  ( -oo  e.  CC  <->  ~P ~P U. CC  e.  CC )
93, 8mtbir 631 . . 3  |-  -. -oo  e.  CC
10 recn 7454 . . 3  |-  ( -oo  e.  RR  -> -oo  e.  CC )
119, 10mto 623 . 2  |-  -. -oo  e.  RR
1211nelir 2353 1  |- -oo  e/  RR
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1438    e/ wnel 2350   _Vcvv 2619   ~Pcpw 3425   U.cuni 3648   CCcc 7327   RRcr 7328   +oocpnf 7498   -oocmnf 7499
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 4343  ax-cnex 7415  ax-resscn 7416
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-nel 2351  df-ral 2364  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-uni 3649  df-pnf 7503  df-mnf 7504
This theorem is referenced by:  renemnf  7515  xrltnr  9219  nltmnf  9227
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