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Theorem nltmnf 9695
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 7920 . . . . . . 7  |- -oo  e/  RR
21neli 2424 . . . . . 6  |-  -. -oo  e.  RR
32intnan 915 . . . . 5  |-  -.  ( A  e.  RR  /\ -oo  e.  RR )
43intnanr 916 . . . 4  |-  -.  (
( A  e.  RR  /\ -oo  e.  RR )  /\  A  <RR -oo )
5 pnfnemnf 7932 . . . . . 6  |- +oo  =/= -oo
65nesymi 2373 . . . . 5  |-  -. -oo  = +oo
76intnan 915 . . . 4  |-  -.  ( A  = -oo  /\ -oo  = +oo )
84, 7pm3.2ni 803 . . 3  |-  -.  (
( ( A  e.  RR  /\ -oo  e.  RR )  /\  A  <RR -oo )  \/  ( A  = -oo  /\ -oo  = +oo ) )
96intnan 915 . . . 4  |-  -.  ( A  e.  RR  /\ -oo  = +oo )
102intnan 915 . . . 4  |-  -.  ( A  = -oo  /\ -oo  e.  RR )
119, 10pm3.2ni 803 . . 3  |-  -.  (
( A  e.  RR  /\ -oo  = +oo )  \/  ( A  = -oo  /\ -oo  e.  RR ) )
128, 11pm3.2ni 803 . 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 7934 . . 3  |- -oo  e.  RR*
14 ltxr 9682 . . 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 422 . 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 665 1  |-  ( A  e.  RR*  ->  -.  A  < -oo )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698    = wceq 1335    e. wcel 2128   class class class wbr 3965   RRcr 7731    <RR cltrr 7736   +oocpnf 7909   -oocmnf 7910   RR*cxr 7911    < clt 7912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4592  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917
This theorem is referenced by:  mnfle  9699  xrltnsym  9700  xrlttr  9702  xrltso  9703  xltnegi  9739  xposdif  9786  qbtwnxr  10157  xrmaxiflemab  11144  xrmaxltsup  11155  xrbdtri  11173  blssioo  12945
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