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Theorem xrltnr 11211
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 11206 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 ltnr 9579 . . 3  |-  ( A  e.  RR  ->  -.  A  <  A )
3 pnfnre 9535 . . . . . . . . . 10  |- +oo  e/  RR
43neli 2786 . . . . . . . . 9  |-  -. +oo  e.  RR
54intnan 905 . . . . . . . 8  |-  -.  ( +oo  e.  RR  /\ +oo  e.  RR )
65intnanr 906 . . . . . . 7  |-  -.  (
( +oo  e.  RR  /\ +oo  e.  RR )  /\ +oo 
<RR +oo )
7 pnfnemnf 11207 . . . . . . . . 9  |- +oo  =/= -oo
87neii 2651 . . . . . . . 8  |-  -. +oo  = -oo
98intnanr 906 . . . . . . 7  |-  -.  ( +oo  = -oo  /\ +oo  = +oo )
106, 9pm3.2ni 850 . . . . . 6  |-  -.  (
( ( +oo  e.  RR  /\ +oo  e.  RR )  /\ +oo  <RR +oo )  \/  ( +oo  = -oo  /\ +oo  = +oo )
)
114intnanr 906 . . . . . . 7  |-  -.  ( +oo  e.  RR  /\ +oo  = +oo )
124intnan 905 . . . . . . 7  |-  -.  ( +oo  = -oo  /\ +oo  e.  RR )
1311, 12pm3.2ni 850 . . . . . 6  |-  -.  (
( +oo  e.  RR  /\ +oo  = +oo )  \/  ( +oo  = -oo  /\ +oo  e.  RR ) )
1410, 13pm3.2ni 850 . . . . 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 11202 . . . . . 6  |- +oo  e.  RR*
16 ltxr 11205 . . . . . 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 4404 . . . . 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 9536 . . . . . . . . . 10  |- -oo  e/  RR
2322neli 2786 . . . . . . . . 9  |-  -. -oo  e.  RR
2423intnan 905 . . . . . . . 8  |-  -.  ( -oo  e.  RR  /\ -oo  e.  RR )
2524intnanr 906 . . . . . . 7  |-  -.  (
( -oo  e.  RR  /\ -oo  e.  RR )  /\ -oo 
<RR -oo )
267nesymi 2724 . . . . . . . 8  |-  -. -oo  = +oo
2726intnan 905 . . . . . . 7  |-  -.  ( -oo  = -oo  /\ -oo  = +oo )
2825, 27pm3.2ni 850 . . . . . 6  |-  -.  (
( ( -oo  e.  RR  /\ -oo  e.  RR )  /\ -oo  <RR -oo )  \/  ( -oo  = -oo  /\ -oo  = +oo )
)
2923intnanr 906 . . . . . . 7  |-  -.  ( -oo  e.  RR  /\ -oo  = +oo )
3023intnan 905 . . . . . . 7  |-  -.  ( -oo  = -oo  /\ -oo  e.  RR )
3129, 30pm3.2ni 850 . . . . . 6  |-  -.  (
( -oo  e.  RR  /\ -oo  = +oo )  \/  ( -oo  = -oo  /\ -oo  e.  RR ) )
3228, 31pm3.2ni 850 . . . . 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 11204 . . . . . 6  |- -oo  e.  RR*
34 ltxr 11205 . . . . . 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 4404 . . . . 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 1282 . 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 964    = wceq 1370    e. wcel 1758   class class class wbr 4399   RRcr 9391    <RR cltrr 9396   +oocpnf 9525   -oocmnf 9526   RR*cxr 9527    < clt 9528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-pre-lttri 9466  ax-pre-lttrn 9467
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-po 4748  df-so 4749  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533
This theorem is referenced by:  xrltnsym  11224  xrlttri  11226  nltpnft  11248  ngtmnft  11249  xrsupsslem  11379  xrinfmsslem  11380  xrub  11384  lbioo  11441  ubioo  11442
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