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Theorem xrltnr 11326
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 11321 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 ltnr 9675 . . 3  |-  ( A  e.  RR  ->  -.  A  <  A )
3 pnfnre 9631 . . . . . . . . . 10  |- +oo  e/  RR
43neli 2802 . . . . . . . . 9  |-  -. +oo  e.  RR
54intnan 912 . . . . . . . 8  |-  -.  ( +oo  e.  RR  /\ +oo  e.  RR )
65intnanr 913 . . . . . . 7  |-  -.  (
( +oo  e.  RR  /\ +oo  e.  RR )  /\ +oo 
<RR +oo )
7 pnfnemnf 11322 . . . . . . . . 9  |- +oo  =/= -oo
87neii 2666 . . . . . . . 8  |-  -. +oo  = -oo
98intnanr 913 . . . . . . 7  |-  -.  ( +oo  = -oo  /\ +oo  = +oo )
106, 9pm3.2ni 852 . . . . . 6  |-  -.  (
( ( +oo  e.  RR  /\ +oo  e.  RR )  /\ +oo  <RR +oo )  \/  ( +oo  = -oo  /\ +oo  = +oo )
)
114intnanr 913 . . . . . . 7  |-  -.  ( +oo  e.  RR  /\ +oo  = +oo )
124intnan 912 . . . . . . 7  |-  -.  ( +oo  = -oo  /\ +oo  e.  RR )
1311, 12pm3.2ni 852 . . . . . 6  |-  -.  (
( +oo  e.  RR  /\ +oo  = +oo )  \/  ( +oo  = -oo  /\ +oo  e.  RR ) )
1410, 13pm3.2ni 852 . . . . 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 11317 . . . . . 6  |- +oo  e.  RR*
16 ltxr 11320 . . . . . 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 672 . . . . 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 299 . . . 4  |-  -. +oo  < +oo
19 breq12 4452 . . . . 5  |-  ( ( A  = +oo  /\  A  = +oo )  ->  ( A  <  A  <-> +oo 
< +oo ) )
2019anidms 645 . . . 4  |-  ( A  = +oo  ->  ( A  <  A  <-> +oo  < +oo ) )
2118, 20mtbiri 303 . . 3  |-  ( A  = +oo  ->  -.  A  <  A )
22 mnfnre 9632 . . . . . . . . . 10  |- -oo  e/  RR
2322neli 2802 . . . . . . . . 9  |-  -. -oo  e.  RR
2423intnan 912 . . . . . . . 8  |-  -.  ( -oo  e.  RR  /\ -oo  e.  RR )
2524intnanr 913 . . . . . . 7  |-  -.  (
( -oo  e.  RR  /\ -oo  e.  RR )  /\ -oo 
<RR -oo )
267nesymi 2740 . . . . . . . 8  |-  -. -oo  = +oo
2726intnan 912 . . . . . . 7  |-  -.  ( -oo  = -oo  /\ -oo  = +oo )
2825, 27pm3.2ni 852 . . . . . 6  |-  -.  (
( ( -oo  e.  RR  /\ -oo  e.  RR )  /\ -oo  <RR -oo )  \/  ( -oo  = -oo  /\ -oo  = +oo )
)
2923intnanr 913 . . . . . . 7  |-  -.  ( -oo  e.  RR  /\ -oo  = +oo )
3023intnan 912 . . . . . . 7  |-  -.  ( -oo  = -oo  /\ -oo  e.  RR )
3129, 30pm3.2ni 852 . . . . . 6  |-  -.  (
( -oo  e.  RR  /\ -oo  = +oo )  \/  ( -oo  = -oo  /\ -oo  e.  RR ) )
3228, 31pm3.2ni 852 . . . . 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 11319 . . . . . 6  |- -oo  e.  RR*
34 ltxr 11320 . . . . . 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 672 . . . . 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 299 . . . 4  |-  -. -oo  < -oo
37 breq12 4452 . . . . 5  |-  ( ( A  = -oo  /\  A  = -oo )  ->  ( A  <  A  <-> -oo 
< -oo ) )
3837anidms 645 . . . 4  |-  ( A  = -oo  ->  ( A  <  A  <-> -oo  < -oo ) )
3936, 38mtbiri 303 . . 3  |-  ( A  = -oo  ->  -.  A  <  A )
402, 21, 393jaoi 1291 . 2  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  -.  A  <  A )
411, 40sylbi 195 1  |-  ( A  e.  RR*  ->  -.  A  <  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 972    = wceq 1379    e. wcel 1767   class class class wbr 4447   RRcr 9487    <RR cltrr 9492   +oocpnf 9621   -oocmnf 9622   RR*cxr 9623    < clt 9624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-pre-lttri 9562  ax-pre-lttrn 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629
This theorem is referenced by:  xrltnsym  11339  xrlttri  11341  nltpnft  11363  ngtmnft  11364  xrsupsslem  11494  xrinfmsslem  11495  xrub  11499  lbioo  11556  ubioo  11557  lbioc  31117
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