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Theorem xrltnsym 8996
Description: Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)
Assertion
Ref Expression
xrltnsym  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  -.  B  <  A ) )

Proof of Theorem xrltnsym
StepHypRef Expression
1 elxr 8980 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 8980 . 2  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 ltnsym 7316 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  -.  B  <  A
) )
4 rexr 7278 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
5 pnfnlt 8990 . . . . . . . 8  |-  ( A  e.  RR*  ->  -. +oo  <  A )
64, 5syl 14 . . . . . . 7  |-  ( A  e.  RR  ->  -. +oo 
<  A )
76adantr 270 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  -. +oo  <  A
)
8 breq1 3808 . . . . . . 7  |-  ( B  = +oo  ->  ( B  <  A  <-> +oo  <  A
) )
98adantl 271 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( B  <  A  <-> +oo 
<  A ) )
107, 9mtbird 631 . . . . 5  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  -.  B  <  A
)
1110a1d 22 . . . 4  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A  <  B  ->  -.  B  <  A
) )
12 nltmnf 8991 . . . . . . . 8  |-  ( A  e.  RR*  ->  -.  A  < -oo )
134, 12syl 14 . . . . . . 7  |-  ( A  e.  RR  ->  -.  A  < -oo )
1413adantr 270 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  A  < -oo )
15 breq2 3809 . . . . . . 7  |-  ( B  = -oo  ->  ( A  <  B  <->  A  < -oo ) )
1615adantl 271 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  <->  A  < -oo ) )
1714, 16mtbird 631 . . . . 5  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  A  <  B
)
1817pm2.21d 582 . . . 4  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  ->  -.  B  <  A
) )
193, 11, 183jaodan 1238 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  -.  B  <  A ) )
20 pnfnlt 8990 . . . . . . 7  |-  ( B  e.  RR*  ->  -. +oo  <  B )
2120adantl 271 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -. +oo  <  B )
22 breq1 3808 . . . . . . 7  |-  ( A  = +oo  ->  ( A  <  B  <-> +oo  <  B
) )
2322adantr 270 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  <-> +oo 
<  B ) )
2421, 23mtbird 631 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -.  A  <  B )
2524pm2.21d 582 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  ->  -.  B  <  A
) )
262, 25sylan2br 282 . . 3  |-  ( ( A  = +oo  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  -.  B  <  A ) )
27 rexr 7278 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
28 nltmnf 8991 . . . . . . . 8  |-  ( B  e.  RR*  ->  -.  B  < -oo )
2927, 28syl 14 . . . . . . 7  |-  ( B  e.  RR  ->  -.  B  < -oo )
3029adantl 271 . . . . . 6  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  -.  B  < -oo )
31 breq2 3809 . . . . . . 7  |-  ( A  = -oo  ->  ( B  <  A  <->  B  < -oo ) )
3231adantr 270 . . . . . 6  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( B  <  A  <->  B  < -oo ) )
3330, 32mtbird 631 . . . . 5  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  -.  B  <  A
)
3433a1d 22 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( A  <  B  ->  -.  B  <  A
) )
35 mnfxr 7289 . . . . . . . 8  |- -oo  e.  RR*
36 pnfnlt 8990 . . . . . . . 8  |-  ( -oo  e.  RR*  ->  -. +oo  < -oo )
3735, 36ax-mp 7 . . . . . . 7  |-  -. +oo  < -oo
38 breq12 3810 . . . . . . 7  |-  ( ( B  = +oo  /\  A  = -oo )  ->  ( B  <  A  <-> +oo 
< -oo ) )
3937, 38mtbiri 633 . . . . . 6  |-  ( ( B  = +oo  /\  A  = -oo )  ->  -.  B  <  A
)
4039ancoms 264 . . . . 5  |-  ( ( A  = -oo  /\  B  = +oo )  ->  -.  B  <  A
)
4140a1d 22 . . . 4  |-  ( ( A  = -oo  /\  B  = +oo )  ->  ( A  <  B  ->  -.  B  <  A
) )
42 xrltnr 8983 . . . . . . 7  |-  ( -oo  e.  RR*  ->  -. -oo  < -oo )
4335, 42ax-mp 7 . . . . . 6  |-  -. -oo  < -oo
44 breq12 3810 . . . . . 6  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( A  <  B  <-> -oo 
< -oo ) )
4543, 44mtbiri 633 . . . . 5  |-  ( ( A  = -oo  /\  B  = -oo )  ->  -.  A  <  B
)
4645pm2.21d 582 . . . 4  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( A  <  B  ->  -.  B  <  A
) )
4734, 41, 463jaodan 1238 . . 3  |-  ( ( A  = -oo  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  -.  B  <  A ) )
4819, 26, 473jaoian 1237 . 2  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  -> 
( A  <  B  ->  -.  B  <  A
) )
491, 2, 48syl2anb 285 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  -.  B  <  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ w3o 919    = wceq 1285    e. wcel 1434   class class class wbr 3805   RRcr 7094   +oocpnf 7264   -oocmnf 7265   RR*cxr 7266    < clt 7267
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-pre-ltirr 7202  ax-pre-lttrn 7204
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-xp 4397  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272
This theorem is referenced by:  xrltnsym2  8997  xrltle  9001
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