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Theorem xrltnr 9219
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 9216 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 ltnr 7541 . . 3  |-  ( A  e.  RR  ->  -.  A  <  A )
3 pnfnre 7508 . . . . . . . . . 10  |- +oo  e/  RR
43neli 2352 . . . . . . . . 9  |-  -. +oo  e.  RR
54intnan 876 . . . . . . . 8  |-  -.  ( +oo  e.  RR  /\ +oo  e.  RR )
65intnanr 877 . . . . . . 7  |-  -.  (
( +oo  e.  RR  /\ +oo  e.  RR )  /\ +oo 
<RR +oo )
7 pnfnemnf 7521 . . . . . . . . 9  |- +oo  =/= -oo
87neii 2257 . . . . . . . 8  |-  -. +oo  = -oo
98intnanr 877 . . . . . . 7  |-  -.  ( +oo  = -oo  /\ +oo  = +oo )
106, 9pm3.2ni 762 . . . . . 6  |-  -.  (
( ( +oo  e.  RR  /\ +oo  e.  RR )  /\ +oo  <RR +oo )  \/  ( +oo  = -oo  /\ +oo  = +oo )
)
114intnanr 877 . . . . . . 7  |-  -.  ( +oo  e.  RR  /\ +oo  = +oo )
124intnan 876 . . . . . . 7  |-  -.  ( +oo  = -oo  /\ +oo  e.  RR )
1311, 12pm3.2ni 762 . . . . . 6  |-  -.  (
( +oo  e.  RR  /\ +oo  = +oo )  \/  ( +oo  = -oo  /\ +oo  e.  RR ) )
1410, 13pm3.2ni 762 . . . . 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 7519 . . . . . 6  |- +oo  e.  RR*
16 ltxr 9215 . . . . . 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 417 . . . . 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 631 . . . 4  |-  -. +oo  < +oo
19 breq12 3842 . . . . 5  |-  ( ( A  = +oo  /\  A  = +oo )  ->  ( A  <  A  <-> +oo 
< +oo ) )
2019anidms 389 . . . 4  |-  ( A  = +oo  ->  ( A  <  A  <-> +oo  < +oo ) )
2118, 20mtbiri 635 . . 3  |-  ( A  = +oo  ->  -.  A  <  A )
22 mnfnre 7509 . . . . . . . . . 10  |- -oo  e/  RR
2322neli 2352 . . . . . . . . 9  |-  -. -oo  e.  RR
2423intnan 876 . . . . . . . 8  |-  -.  ( -oo  e.  RR  /\ -oo  e.  RR )
2524intnanr 877 . . . . . . 7  |-  -.  (
( -oo  e.  RR  /\ -oo  e.  RR )  /\ -oo 
<RR -oo )
267nesymi 2301 . . . . . . . 8  |-  -. -oo  = +oo
2726intnan 876 . . . . . . 7  |-  -.  ( -oo  = -oo  /\ -oo  = +oo )
2825, 27pm3.2ni 762 . . . . . 6  |-  -.  (
( ( -oo  e.  RR  /\ -oo  e.  RR )  /\ -oo  <RR -oo )  \/  ( -oo  = -oo  /\ -oo  = +oo )
)
2923intnanr 877 . . . . . . 7  |-  -.  ( -oo  e.  RR  /\ -oo  = +oo )
3023intnan 876 . . . . . . 7  |-  -.  ( -oo  = -oo  /\ -oo  e.  RR )
3129, 30pm3.2ni 762 . . . . . 6  |-  -.  (
( -oo  e.  RR  /\ -oo  = +oo )  \/  ( -oo  = -oo  /\ -oo  e.  RR ) )
3228, 31pm3.2ni 762 . . . . 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 7523 . . . . . 6  |- -oo  e.  RR*
34 ltxr 9215 . . . . . 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 417 . . . . 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 631 . . . 4  |-  -. -oo  < -oo
37 breq12 3842 . . . . 5  |-  ( ( A  = -oo  /\  A  = -oo )  ->  ( A  <  A  <-> -oo 
< -oo ) )
3837anidms 389 . . . 4  |-  ( A  = -oo  ->  ( A  <  A  <-> -oo  < -oo ) )
3936, 38mtbiri 635 . . 3  |-  ( A  = -oo  ->  -.  A  <  A )
402, 21, 393jaoi 1239 . 2  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  -.  A  <  A )
411, 40sylbi 119 1  |-  ( A  e.  RR*  ->  -.  A  <  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    \/ w3o 923    = wceq 1289    e. wcel 1438   class class class wbr 3837   RRcr 7328    <RR cltrr 7333   +oocpnf 7498   -oocmnf 7499   RR*cxr 7500    < clt 7501
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-pre-ltirr 7436
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-xp 4434  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506
This theorem is referenced by:  xrltnsym  9232  xrltso  9235  xrlttri3  9236  xrleid  9238  xrltne  9247  nltpnft  9248  ngtmnft  9249  xrrebnd  9250  lbioog  9300  ubioog  9301
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