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Theorem xrltnr 9517
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 9514 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 ltnr 7805 . . 3  |-  ( A  e.  RR  ->  -.  A  <  A )
3 pnfnre 7771 . . . . . . . . . 10  |- +oo  e/  RR
43neli 2380 . . . . . . . . 9  |-  -. +oo  e.  RR
54intnan 897 . . . . . . . 8  |-  -.  ( +oo  e.  RR  /\ +oo  e.  RR )
65intnanr 898 . . . . . . 7  |-  -.  (
( +oo  e.  RR  /\ +oo  e.  RR )  /\ +oo 
<RR +oo )
7 pnfnemnf 7784 . . . . . . . . 9  |- +oo  =/= -oo
87neii 2285 . . . . . . . 8  |-  -. +oo  = -oo
98intnanr 898 . . . . . . 7  |-  -.  ( +oo  = -oo  /\ +oo  = +oo )
106, 9pm3.2ni 785 . . . . . 6  |-  -.  (
( ( +oo  e.  RR  /\ +oo  e.  RR )  /\ +oo  <RR +oo )  \/  ( +oo  = -oo  /\ +oo  = +oo )
)
114intnanr 898 . . . . . . 7  |-  -.  ( +oo  e.  RR  /\ +oo  = +oo )
124intnan 897 . . . . . . 7  |-  -.  ( +oo  = -oo  /\ +oo  e.  RR )
1311, 12pm3.2ni 785 . . . . . 6  |-  -.  (
( +oo  e.  RR  /\ +oo  = +oo )  \/  ( +oo  = -oo  /\ +oo  e.  RR ) )
1410, 13pm3.2ni 785 . . . . 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 7782 . . . . . 6  |- +oo  e.  RR*
16 ltxr 9513 . . . . . 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 420 . . . . 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 643 . . . 4  |-  -. +oo  < +oo
19 breq12 3902 . . . . 5  |-  ( ( A  = +oo  /\  A  = +oo )  ->  ( A  <  A  <-> +oo 
< +oo ) )
2019anidms 392 . . . 4  |-  ( A  = +oo  ->  ( A  <  A  <-> +oo  < +oo ) )
2118, 20mtbiri 647 . . 3  |-  ( A  = +oo  ->  -.  A  <  A )
22 mnfnre 7772 . . . . . . . . . 10  |- -oo  e/  RR
2322neli 2380 . . . . . . . . 9  |-  -. -oo  e.  RR
2423intnan 897 . . . . . . . 8  |-  -.  ( -oo  e.  RR  /\ -oo  e.  RR )
2524intnanr 898 . . . . . . 7  |-  -.  (
( -oo  e.  RR  /\ -oo  e.  RR )  /\ -oo 
<RR -oo )
267nesymi 2329 . . . . . . . 8  |-  -. -oo  = +oo
2726intnan 897 . . . . . . 7  |-  -.  ( -oo  = -oo  /\ -oo  = +oo )
2825, 27pm3.2ni 785 . . . . . 6  |-  -.  (
( ( -oo  e.  RR  /\ -oo  e.  RR )  /\ -oo  <RR -oo )  \/  ( -oo  = -oo  /\ -oo  = +oo )
)
2923intnanr 898 . . . . . . 7  |-  -.  ( -oo  e.  RR  /\ -oo  = +oo )
3023intnan 897 . . . . . . 7  |-  -.  ( -oo  = -oo  /\ -oo  e.  RR )
3129, 30pm3.2ni 785 . . . . . 6  |-  -.  (
( -oo  e.  RR  /\ -oo  = +oo )  \/  ( -oo  = -oo  /\ -oo  e.  RR ) )
3228, 31pm3.2ni 785 . . . . 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 7786 . . . . . 6  |- -oo  e.  RR*
34 ltxr 9513 . . . . . 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 420 . . . . 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 643 . . . 4  |-  -. -oo  < -oo
37 breq12 3902 . . . . 5  |-  ( ( A  = -oo  /\  A  = -oo )  ->  ( A  <  A  <-> -oo 
< -oo ) )
3837anidms 392 . . . 4  |-  ( A  = -oo  ->  ( A  <  A  <-> -oo  < -oo ) )
3936, 38mtbiri 647 . . 3  |-  ( A  = -oo  ->  -.  A  <  A )
402, 21, 393jaoi 1264 . 2  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  -.  A  <  A )
411, 40sylbi 120 1  |-  ( A  e.  RR*  ->  -.  A  <  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680    \/ w3o 944    = wceq 1314    e. wcel 1463   class class class wbr 3897   RRcr 7583    <RR cltrr 7588   +oocpnf 7761   -oocmnf 7762   RR*cxr 7763    < clt 7764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-pre-ltirr 7696
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-xp 4513  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769
This theorem is referenced by:  xrltnsym  9530  xrltso  9533  xrlttri3  9534  xrleid  9537  xrltne  9547  nltpnft  9548  ngtmnft  9551  xrrebnd  9553  xposdif  9616  lbioog  9647  ubioog  9648  xrmaxleim  10964  xrmaxiflemlub  10968
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