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Theorem nltmnf 11212
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 9529 . . . . . . 7  |- -oo  e/  RR
21neli 2783 . . . . . 6  |-  -. -oo  e.  RR
32intnan 905 . . . . 5  |-  -.  ( A  e.  RR  /\ -oo  e.  RR )
43intnanr 906 . . . 4  |-  -.  (
( A  e.  RR  /\ -oo  e.  RR )  /\  A  <RR -oo )
5 pnfnemnf 11200 . . . . . 6  |- +oo  =/= -oo
65nesymi 2721 . . . . 5  |-  -. -oo  = +oo
76intnan 905 . . . 4  |-  -.  ( A  = -oo  /\ -oo  = +oo )
84, 7pm3.2ni 850 . . 3  |-  -.  (
( ( A  e.  RR  /\ -oo  e.  RR )  /\  A  <RR -oo )  \/  ( A  = -oo  /\ -oo  = +oo ) )
96intnan 905 . . . 4  |-  -.  ( A  e.  RR  /\ -oo  = +oo )
102intnan 905 . . . 4  |-  -.  ( A  = -oo  /\ -oo  e.  RR )
119, 10pm3.2ni 850 . . 3  |-  -.  (
( A  e.  RR  /\ -oo  = +oo )  \/  ( A  = -oo  /\ -oo  e.  RR ) )
128, 11pm3.2ni 850 . 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 11197 . . 3  |- -oo  e.  RR*
14 ltxr 11198 . . 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 671 . 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 303 1  |-  ( A  e.  RR*  ->  -.  A  < -oo )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4392   RRcr 9384    <RR cltrr 9389   +oocpnf 9518   -oocmnf 9519   RR*cxr 9520    < clt 9521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526
This theorem is referenced by:  mnfle  11216  xrltnsym  11217  xrlttr  11220  qbtwnxr  11273  xltnegi  11289  xmullem2  11331  xmulasslem2  11348  xlemul1a  11354  xrsupexmnf  11370  xrsupsslem  11372  xrinfmsslem  11373  xrsup0  11389  mnfnei  18943  blssioo  20490  deg1add  21693
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