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Theorem nltmnf 8709
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 7068 . . . . . . 7  |- -oo  e/  RR
21neli 2299 . . . . . 6  |-  -. -oo  e.  RR
32intnan 838 . . . . 5  |-  -.  ( A  e.  RR  /\ -oo  e.  RR )
43intnanr 839 . . . 4  |-  -.  (
( A  e.  RR  /\ -oo  e.  RR )  /\  A  <RR -oo )
5 pnfnemnf 8697 . . . . . 6  |- +oo  =/= -oo
65nesymi 2251 . . . . 5  |-  -. -oo  = +oo
76intnan 838 . . . 4  |-  -.  ( A  = -oo  /\ -oo  = +oo )
84, 7pm3.2ni 726 . . 3  |-  -.  (
( ( A  e.  RR  /\ -oo  e.  RR )  /\  A  <RR -oo )  \/  ( A  = -oo  /\ -oo  = +oo ) )
96intnan 838 . . . 4  |-  -.  ( A  e.  RR  /\ -oo  = +oo )
102intnan 838 . . . 4  |-  -.  ( A  = -oo  /\ -oo  e.  RR )
119, 10pm3.2ni 726 . . 3  |-  -.  (
( A  e.  RR  /\ -oo  = +oo )  \/  ( A  = -oo  /\ -oo  e.  RR ) )
128, 11pm3.2ni 726 . 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 8694 . . 3  |- -oo  e.  RR*
14 ltxr 8695 . . 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 401 . 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 600 1  |-  ( A  e.  RR*  ->  -.  A  < -oo )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 97    <-> wb 98    \/ wo 629    = wceq 1243    e. wcel 1393   class class class wbr 3764   RRcr 6888    <RR cltrr 6893   +oocpnf 7057   -oocmnf 7058   RR*cxr 7059    < clt 7060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065
This theorem is referenced by:  mnfle  8713  xrltnsym  8714  xrlttr  8716  xrltso  8717  xltnegi  8748
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