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Theorem nltmnf 8928
Description: No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)
Assertion
Ref Expression
nltmnf  |-  ( A  e.  RR*  ->  -.  A  < -oo )

Proof of Theorem nltmnf
StepHypRef Expression
1 mnfnre 7212 . . . . . . 7  |- -oo  e/  RR
21neli 2342 . . . . . 6  |-  -. -oo  e.  RR
32intnan 872 . . . . 5  |-  -.  ( A  e.  RR  /\ -oo  e.  RR )
43intnanr 873 . . . 4  |-  -.  (
( A  e.  RR  /\ -oo  e.  RR )  /\  A  <RR -oo )
5 pnfnemnf 7224 . . . . . 6  |- +oo  =/= -oo
65nesymi 2292 . . . . 5  |-  -. -oo  = +oo
76intnan 872 . . . 4  |-  -.  ( A  = -oo  /\ -oo  = +oo )
84, 7pm3.2ni 760 . . 3  |-  -.  (
( ( A  e.  RR  /\ -oo  e.  RR )  /\  A  <RR -oo )  \/  ( A  = -oo  /\ -oo  = +oo ) )
96intnan 872 . . . 4  |-  -.  ( A  e.  RR  /\ -oo  = +oo )
102intnan 872 . . . 4  |-  -.  ( A  = -oo  /\ -oo  e.  RR )
119, 10pm3.2ni 760 . . 3  |-  -.  (
( A  e.  RR  /\ -oo  = +oo )  \/  ( A  = -oo  /\ -oo  e.  RR ) )
128, 11pm3.2ni 760 . 2  |-  -.  (
( ( ( A  e.  RR  /\ -oo  e.  RR )  /\  A  <RR -oo )  \/  ( A  = -oo  /\ -oo  = +oo ) )  \/  ( ( A  e.  RR  /\ -oo  = +oo )  \/  ( A  = -oo  /\ -oo  e.  RR ) ) )
13 mnfxr 7226 . . 3  |- -oo  e.  RR*
14 ltxr 8916 . . 3  |-  ( ( A  e.  RR*  /\ -oo  e.  RR* )  ->  ( A  < -oo  <->  ( ( ( ( A  e.  RR  /\ -oo  e.  RR )  /\  A  <RR -oo )  \/  ( A  = -oo  /\ -oo  = +oo ) )  \/  ( ( A  e.  RR  /\ -oo  = +oo )  \/  ( A  = -oo  /\ -oo  e.  RR ) ) ) ) )
1513, 14mpan2 416 . 2  |-  ( A  e.  RR*  ->  ( A  < -oo  <->  ( ( ( ( A  e.  RR  /\ -oo  e.  RR )  /\  A  <RR -oo )  \/  ( A  = -oo  /\ -oo  = +oo ) )  \/  ( ( A  e.  RR  /\ -oo  = +oo )  \/  ( A  = -oo  /\ -oo  e.  RR ) ) ) ) )
1612, 15mtbiri 633 1  |-  ( A  e.  RR*  ->  -.  A  < -oo )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    = wceq 1285    e. wcel 1434   class class class wbr 3787   RRcr 7031    <RR cltrr 7036   +oocpnf 7201   -oocmnf 7202   RR*cxr 7203    < clt 7204
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-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-cnex 7118  ax-resscn 7119
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-xp 4371  df-pnf 7206  df-mnf 7207  df-xr 7208  df-ltxr 7209
This theorem is referenced by:  mnfle  8932  xrltnsym  8933  xrlttr  8935  xrltso  8936  xltnegi  8967  qbtwnxr  9333
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