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Theorem xrltnsym 9519
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 9503 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 9503 . 2  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 ltnsym 7814 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  -.  B  <  A
) )
4 rexr 7775 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
5 pnfnlt 9513 . . . . . . . 8  |-  ( A  e.  RR*  ->  -. +oo  <  A )
64, 5syl 14 . . . . . . 7  |-  ( A  e.  RR  ->  -. +oo 
<  A )
76adantr 272 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  -. +oo  <  A
)
8 breq1 3900 . . . . . . 7  |-  ( B  = +oo  ->  ( B  <  A  <-> +oo  <  A
) )
98adantl 273 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( B  <  A  <-> +oo 
<  A ) )
107, 9mtbird 645 . . . . 5  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  -.  B  <  A
)
1110a1d 22 . . . 4  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A  <  B  ->  -.  B  <  A
) )
12 nltmnf 9514 . . . . . . . 8  |-  ( A  e.  RR*  ->  -.  A  < -oo )
134, 12syl 14 . . . . . . 7  |-  ( A  e.  RR  ->  -.  A  < -oo )
1413adantr 272 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  A  < -oo )
15 breq2 3901 . . . . . . 7  |-  ( B  = -oo  ->  ( A  <  B  <->  A  < -oo ) )
1615adantl 273 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  <->  A  < -oo ) )
1714, 16mtbird 645 . . . . 5  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  A  <  B
)
1817pm2.21d 591 . . . 4  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  ->  -.  B  <  A
) )
193, 11, 183jaodan 1267 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  -.  B  <  A ) )
20 pnfnlt 9513 . . . . . . 7  |-  ( B  e.  RR*  ->  -. +oo  <  B )
2120adantl 273 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -. +oo  <  B )
22 breq1 3900 . . . . . . 7  |-  ( A  = +oo  ->  ( A  <  B  <-> +oo  <  B
) )
2322adantr 272 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  <-> +oo 
<  B ) )
2421, 23mtbird 645 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -.  A  <  B )
2524pm2.21d 591 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  ->  -.  B  <  A
) )
262, 25sylan2br 284 . . 3  |-  ( ( A  = +oo  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  -.  B  <  A ) )
27 rexr 7775 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
28 nltmnf 9514 . . . . . . . 8  |-  ( B  e.  RR*  ->  -.  B  < -oo )
2927, 28syl 14 . . . . . . 7  |-  ( B  e.  RR  ->  -.  B  < -oo )
3029adantl 273 . . . . . 6  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  -.  B  < -oo )
31 breq2 3901 . . . . . . 7  |-  ( A  = -oo  ->  ( B  <  A  <->  B  < -oo ) )
3231adantr 272 . . . . . 6  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( B  <  A  <->  B  < -oo ) )
3330, 32mtbird 645 . . . . 5  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  -.  B  <  A
)
3433a1d 22 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( A  <  B  ->  -.  B  <  A
) )
35 mnfxr 7786 . . . . . . . 8  |- -oo  e.  RR*
36 pnfnlt 9513 . . . . . . . 8  |-  ( -oo  e.  RR*  ->  -. +oo  < -oo )
3735, 36ax-mp 5 . . . . . . 7  |-  -. +oo  < -oo
38 breq12 3902 . . . . . . 7  |-  ( ( B  = +oo  /\  A  = -oo )  ->  ( B  <  A  <-> +oo 
< -oo ) )
3937, 38mtbiri 647 . . . . . 6  |-  ( ( B  = +oo  /\  A  = -oo )  ->  -.  B  <  A
)
4039ancoms 266 . . . . 5  |-  ( ( A  = -oo  /\  B  = +oo )  ->  -.  B  <  A
)
4140a1d 22 . . . 4  |-  ( ( A  = -oo  /\  B  = +oo )  ->  ( A  <  B  ->  -.  B  <  A
) )
42 xrltnr 9506 . . . . . . 7  |-  ( -oo  e.  RR*  ->  -. -oo  < -oo )
4335, 42ax-mp 5 . . . . . 6  |-  -. -oo  < -oo
44 breq12 3902 . . . . . 6  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( A  <  B  <-> -oo 
< -oo ) )
4543, 44mtbiri 647 . . . . 5  |-  ( ( A  = -oo  /\  B  = -oo )  ->  -.  A  <  B
)
4645pm2.21d 591 . . . 4  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( A  <  B  ->  -.  B  <  A
) )
4734, 41, 463jaodan 1267 . . 3  |-  ( ( A  = -oo  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  -.  B  <  A ) )
4819, 26, 473jaoian 1266 . 2  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  -> 
( A  <  B  ->  -.  B  <  A
) )
491, 2, 48syl2anb 287 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  -.  B  <  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ w3o 944    = wceq 1314    e. wcel 1463   class class class wbr 3897   RRcr 7583   +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  ax-pre-lttrn 7698
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:  xrltnsym2  9520  xrltle  9524
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