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Theorem nltmnf 10651
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 9054 . . . . . . 7  |-  -oo  e/  RR
2 df-nel 2546 . . . . . . 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 10642 . . . . . . 7  |-  +oo  =/=  -oo
76necomi 2625 . . . . . 6  |-  -oo  =/=  +oo
8 df-ne 2545 . . . . . 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 10639 . . 3  |-  -oo  e.  RR*
17 ltxr 10640 . . 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 1649    e. wcel 1717    =/= wne 2543    e/ wnel 2544   class class class wbr 4146   RRcr 8915    <RR cltrr 8920    +oocpnf 9043    -oocmnf 9044   RR*cxr 9045    < clt 9046
This theorem is referenced by:  mnfle  10654  xrltnsym  10655  xrlttr  10658  qbtwnxr  10711  xltnegi  10727  xmullem2  10769  xmulasslem2  10786  xlemul1a  10792  xrsupexmnf  10808  xrsupsslem  10810  xrinfmsslem  10811  xrsup0  10827  mnfnei  17200  blssioo  18690  deg1add  19886
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051
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