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Theorem nltmnf 10484
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 8891 . . . . . . 7  |-  -oo  e/  RR
2 df-nel 2462 . . . . . . 7  |-  (  -oo  e/  RR  <->  -.  -oo  e.  RR )
31, 2mpbi 199 . . . . . 6  |-  -.  -oo  e.  RR
43intnan 880 . . . . 5  |-  -.  ( A  e.  RR  /\  -oo  e.  RR )
54intnanr 881 . . . 4  |-  -.  (
( A  e.  RR  /\ 
-oo  e.  RR )  /\  A  <RR  -oo )
6 pnfnemnf 10475 . . . . . . 7  |-  +oo  =/=  -oo
76necomi 2541 . . . . . 6  |-  -oo  =/=  +oo
8 df-ne 2461 . . . . . 6  |-  (  -oo  =/=  +oo  <->  -.  -oo  =  +oo )
97, 8mpbi 199 . . . . 5  |-  -.  -oo  =  +oo
109intnan 880 . . . 4  |-  -.  ( A  =  -oo  /\  -oo  =  +oo )
115, 10pm3.2ni 827 . . 3  |-  -.  (
( ( A  e.  RR  /\  -oo  e.  RR )  /\  A  <RR  -oo )  \/  ( A  =  -oo  /\  -oo  =  +oo ) )
129intnan 880 . . . 4  |-  -.  ( A  e.  RR  /\  -oo  =  +oo )
133intnan 880 . . . 4  |-  -.  ( A  =  -oo  /\  -oo  e.  RR )
1412, 13pm3.2ni 827 . . 3  |-  -.  (
( A  e.  RR  /\ 
-oo  =  +oo )  \/  ( A  =  -oo  /\ 
-oo  e.  RR )
)
1511, 14pm3.2ni 827 . 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 10472 . . 3  |-  -oo  e.  RR*
17 ltxr 10473 . . 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 652 . 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 294 1  |-  ( A  e.  RR*  ->  -.  A  <  -oo )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    e/ wnel 2460   class class class wbr 4039   RRcr 8752    <RR cltrr 8757    +oocpnf 8880    -oocmnf 8881   RR*cxr 8882    < clt 8883
This theorem is referenced by:  mnfle  10486  xrltnsym  10487  xrlttr  10490  qbtwnxr  10543  xltnegi  10559  xmullem2  10601  xmulasslem2  10618  xlemul1a  10624  xrsupexmnf  10639  xrsupsslem  10641  xrinfmsslem  10642  xrsup0  10658  mnfnei  16967  blssioo  18317  deg1add  19505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888
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