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Theorem nltmnf 10718
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 9120 . . . . . . 7  |-  -oo  e/  RR
2 df-nel 2601 . . . . . . 7  |-  (  -oo  e/  RR  <->  -.  -oo  e.  RR )
31, 2mpbi 200 . . . . . 6  |-  -.  -oo  e.  RR
43intnan 881 . . . . 5  |-  -.  ( A  e.  RR  /\  -oo  e.  RR )
54intnanr 882 . . . 4  |-  -.  (
( A  e.  RR  /\ 
-oo  e.  RR )  /\  A  <RR  -oo )
6 pnfnemnf 10709 . . . . . . 7  |-  +oo  =/=  -oo
76necomi 2680 . . . . . 6  |-  -oo  =/=  +oo
8 df-ne 2600 . . . . . 6  |-  (  -oo  =/=  +oo  <->  -.  -oo  =  +oo )
97, 8mpbi 200 . . . . 5  |-  -.  -oo  =  +oo
109intnan 881 . . . 4  |-  -.  ( A  =  -oo  /\  -oo  =  +oo )
115, 10pm3.2ni 828 . . 3  |-  -.  (
( ( A  e.  RR  /\  -oo  e.  RR )  /\  A  <RR  -oo )  \/  ( A  =  -oo  /\  -oo  =  +oo ) )
129intnan 881 . . . 4  |-  -.  ( A  e.  RR  /\  -oo  =  +oo )
133intnan 881 . . . 4  |-  -.  ( A  =  -oo  /\  -oo  e.  RR )
1412, 13pm3.2ni 828 . . 3  |-  -.  (
( A  e.  RR  /\ 
-oo  =  +oo )  \/  ( A  =  -oo  /\ 
-oo  e.  RR )
)
1511, 14pm3.2ni 828 . 2  |-  -.  (
( ( ( A  e.  RR  /\  -oo  e.  RR )  /\  A  <RR 
-oo )  \/  ( A  =  -oo  /\  -oo  =  +oo ) )  \/  ( ( A  e.  RR  /\  -oo  =  +oo )  \/  ( A  =  -oo  /\  -oo  e.  RR ) ) )
16 mnfxr 10706 . . 3  |-  -oo  e.  RR*
17 ltxr 10707 . . 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 )
) ) ) )
1816, 17mpan2 653 . 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 )
) ) ) )
1915, 18mtbiri 295 1  |-  ( A  e.  RR*  ->  -.  A  <  -oo )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598    e/ wnel 2599   class class class wbr 4204   RRcr 8981    <RR cltrr 8986    +oocpnf 9109    -oocmnf 9110   RR*cxr 9111    < clt 9112
This theorem is referenced by:  mnfle  10721  xrltnsym  10722  xrlttr  10725  qbtwnxr  10778  xltnegi  10794  xmullem2  10836  xmulasslem2  10853  xlemul1a  10859  xrsupexmnf  10875  xrsupsslem  10877  xrinfmsslem  10878  xrsup0  10894  mnfnei  17277  blssioo  18818  deg1add  20018
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117
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