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Theorem mnfnre 7223
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 7159 . . . . 5  |-  CC  e.  _V
2 2pwuninelg 5932 . . . . 5  |-  ( CC  e.  _V  ->  -.  ~P ~P U. CC  e.  CC )
31, 2ax-mp 7 . . . 4  |-  -.  ~P ~P U. CC  e.  CC
4 df-mnf 7218 . . . . . 6  |- -oo  =  ~P +oo
5 df-pnf 7217 . . . . . . 7  |- +oo  =  ~P U. CC
65pweqi 3394 . . . . . 6  |-  ~P +oo  =  ~P ~P U. CC
74, 6eqtri 2102 . . . . 5  |- -oo  =  ~P ~P U. CC
87eleq1i 2145 . . . 4  |-  ( -oo  e.  CC  <->  ~P ~P U. CC  e.  CC )
93, 8mtbir 629 . . 3  |-  -. -oo  e.  CC
10 recn 7168 . . 3  |-  ( -oo  e.  RR  -> -oo  e.  CC )
119, 10mto 621 . 2  |-  -. -oo  e.  RR
1211nelir 2343 1  |- -oo  e/  RR
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1434    e/ wnel 2340   _Vcvv 2602   ~Pcpw 3390   U.cuni 3609   CCcc 7041   RRcr 7042   +oocpnf 7212   -oocmnf 7213
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 2064  ax-setind 4288  ax-cnex 7129  ax-resscn 7130
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-nel 2341  df-ral 2354  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-uni 3610  df-pnf 7217  df-mnf 7218
This theorem is referenced by:  renemnf  7229  xrltnr  8931  nltmnf  8939
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