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Theorem xrltnr 9779
Description: The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)
Assertion
Ref Expression
xrltnr  |-  ( A  e.  RR*  ->  -.  A  <  A )

Proof of Theorem xrltnr
StepHypRef Expression
1 elxr 9776 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 ltnr 8034 . . 3  |-  ( A  e.  RR  ->  -.  A  <  A )
3 pnfnre 7999 . . . . . . . . . 10  |- +oo  e/  RR
43neli 2444 . . . . . . . . 9  |-  -. +oo  e.  RR
54intnan 929 . . . . . . . 8  |-  -.  ( +oo  e.  RR  /\ +oo  e.  RR )
65intnanr 930 . . . . . . 7  |-  -.  (
( +oo  e.  RR  /\ +oo  e.  RR )  /\ +oo 
<RR +oo )
7 pnfnemnf 8012 . . . . . . . . 9  |- +oo  =/= -oo
87neii 2349 . . . . . . . 8  |-  -. +oo  = -oo
98intnanr 930 . . . . . . 7  |-  -.  ( +oo  = -oo  /\ +oo  = +oo )
106, 9pm3.2ni 813 . . . . . 6  |-  -.  (
( ( +oo  e.  RR  /\ +oo  e.  RR )  /\ +oo  <RR +oo )  \/  ( +oo  = -oo  /\ +oo  = +oo )
)
114intnanr 930 . . . . . . 7  |-  -.  ( +oo  e.  RR  /\ +oo  = +oo )
124intnan 929 . . . . . . 7  |-  -.  ( +oo  = -oo  /\ +oo  e.  RR )
1311, 12pm3.2ni 813 . . . . . 6  |-  -.  (
( +oo  e.  RR  /\ +oo  = +oo )  \/  ( +oo  = -oo  /\ +oo  e.  RR ) )
1410, 13pm3.2ni 813 . . . . 5  |-  -.  (
( ( ( +oo  e.  RR  /\ +oo  e.  RR )  /\ +oo  <RR +oo )  \/  ( +oo  = -oo  /\ +oo  = +oo ) )  \/  (
( +oo  e.  RR  /\ +oo  = +oo )  \/  ( +oo  = -oo  /\ +oo  e.  RR ) ) )
15 pnfxr 8010 . . . . . 6  |- +oo  e.  RR*
16 ltxr 9775 . . . . . 6  |-  ( ( +oo  e.  RR*  /\ +oo  e.  RR* )  ->  ( +oo  < +oo  <->  ( ( ( ( +oo  e.  RR  /\ +oo  e.  RR )  /\ +oo 
<RR +oo )  \/  ( +oo  = -oo  /\ +oo  = +oo ) )  \/  ( ( +oo  e.  RR  /\ +oo  = +oo )  \/  ( +oo  = -oo  /\ +oo  e.  RR ) ) ) ) )
1715, 15, 16mp2an 426 . . . . 5  |-  ( +oo  < +oo  <->  ( ( ( ( +oo  e.  RR  /\ +oo  e.  RR )  /\ +oo 
<RR +oo )  \/  ( +oo  = -oo  /\ +oo  = +oo ) )  \/  ( ( +oo  e.  RR  /\ +oo  = +oo )  \/  ( +oo  = -oo  /\ +oo  e.  RR ) ) ) )
1814, 17mtbir 671 . . . 4  |-  -. +oo  < +oo
19 breq12 4009 . . . . 5  |-  ( ( A  = +oo  /\  A  = +oo )  ->  ( A  <  A  <-> +oo 
< +oo ) )
2019anidms 397 . . . 4  |-  ( A  = +oo  ->  ( A  <  A  <-> +oo  < +oo ) )
2118, 20mtbiri 675 . . 3  |-  ( A  = +oo  ->  -.  A  <  A )
22 mnfnre 8000 . . . . . . . . . 10  |- -oo  e/  RR
2322neli 2444 . . . . . . . . 9  |-  -. -oo  e.  RR
2423intnan 929 . . . . . . . 8  |-  -.  ( -oo  e.  RR  /\ -oo  e.  RR )
2524intnanr 930 . . . . . . 7  |-  -.  (
( -oo  e.  RR  /\ -oo  e.  RR )  /\ -oo 
<RR -oo )
267nesymi 2393 . . . . . . . 8  |-  -. -oo  = +oo
2726intnan 929 . . . . . . 7  |-  -.  ( -oo  = -oo  /\ -oo  = +oo )
2825, 27pm3.2ni 813 . . . . . 6  |-  -.  (
( ( -oo  e.  RR  /\ -oo  e.  RR )  /\ -oo  <RR -oo )  \/  ( -oo  = -oo  /\ -oo  = +oo )
)
2923intnanr 930 . . . . . . 7  |-  -.  ( -oo  e.  RR  /\ -oo  = +oo )
3023intnan 929 . . . . . . 7  |-  -.  ( -oo  = -oo  /\ -oo  e.  RR )
3129, 30pm3.2ni 813 . . . . . 6  |-  -.  (
( -oo  e.  RR  /\ -oo  = +oo )  \/  ( -oo  = -oo  /\ -oo  e.  RR ) )
3228, 31pm3.2ni 813 . . . . 5  |-  -.  (
( ( ( -oo  e.  RR  /\ -oo  e.  RR )  /\ -oo  <RR -oo )  \/  ( -oo  = -oo  /\ -oo  = +oo ) )  \/  (
( -oo  e.  RR  /\ -oo  = +oo )  \/  ( -oo  = -oo  /\ -oo  e.  RR ) ) )
33 mnfxr 8014 . . . . . 6  |- -oo  e.  RR*
34 ltxr 9775 . . . . . 6  |-  ( ( -oo  e.  RR*  /\ -oo  e.  RR* )  ->  ( -oo  < -oo  <->  ( ( ( ( -oo  e.  RR  /\ -oo  e.  RR )  /\ -oo 
<RR -oo )  \/  ( -oo  = -oo  /\ -oo  = +oo ) )  \/  ( ( -oo  e.  RR  /\ -oo  = +oo )  \/  ( -oo  = -oo  /\ -oo  e.  RR ) ) ) ) )
3533, 33, 34mp2an 426 . . . . 5  |-  ( -oo  < -oo  <->  ( ( ( ( -oo  e.  RR  /\ -oo  e.  RR )  /\ -oo 
<RR -oo )  \/  ( -oo  = -oo  /\ -oo  = +oo ) )  \/  ( ( -oo  e.  RR  /\ -oo  = +oo )  \/  ( -oo  = -oo  /\ -oo  e.  RR ) ) ) )
3632, 35mtbir 671 . . . 4  |-  -. -oo  < -oo
37 breq12 4009 . . . . 5  |-  ( ( A  = -oo  /\  A  = -oo )  ->  ( A  <  A  <-> -oo 
< -oo ) )
3837anidms 397 . . . 4  |-  ( A  = -oo  ->  ( A  <  A  <-> -oo  < -oo ) )
3936, 38mtbiri 675 . . 3  |-  ( A  = -oo  ->  -.  A  <  A )
402, 21, 393jaoi 1303 . 2  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  -.  A  <  A )
411, 40sylbi 121 1  |-  ( A  e.  RR*  ->  -.  A  <  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708    \/ w3o 977    = wceq 1353    e. wcel 2148   class class class wbr 4004   RRcr 7810    <RR cltrr 7815   +oocpnf 7989   -oocmnf 7990   RR*cxr 7991    < clt 7992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-pre-ltirr 7923
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-xp 4633  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997
This theorem is referenced by:  xrltnsym  9793  xrltso  9796  xrlttri3  9797  xrleid  9800  xrltne  9813  nltpnft  9814  ngtmnft  9817  xrrebnd  9819  xposdif  9882  lbioog  9913  ubioog  9914  xrmaxleim  11252  xrmaxiflemlub  11256
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