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Theorem xrltnsym 9701
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 9684 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 9684 . 2  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 ltnsym 7964 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  -.  B  <  A
) )
4 rexr 7924 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
5 pnfnlt 9695 . . . . . . . 8  |-  ( A  e.  RR*  ->  -. +oo  <  A )
64, 5syl 14 . . . . . . 7  |-  ( A  e.  RR  ->  -. +oo 
<  A )
76adantr 274 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  -. +oo  <  A
)
8 breq1 3969 . . . . . . 7  |-  ( B  = +oo  ->  ( B  <  A  <-> +oo  <  A
) )
98adantl 275 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( B  <  A  <-> +oo 
<  A ) )
107, 9mtbird 663 . . . . 5  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  -.  B  <  A
)
1110a1d 22 . . . 4  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A  <  B  ->  -.  B  <  A
) )
12 nltmnf 9696 . . . . . . . 8  |-  ( A  e.  RR*  ->  -.  A  < -oo )
134, 12syl 14 . . . . . . 7  |-  ( A  e.  RR  ->  -.  A  < -oo )
1413adantr 274 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  A  < -oo )
15 breq2 3970 . . . . . . 7  |-  ( B  = -oo  ->  ( A  <  B  <->  A  < -oo ) )
1615adantl 275 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  <->  A  < -oo ) )
1714, 16mtbird 663 . . . . 5  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  A  <  B
)
1817pm2.21d 609 . . . 4  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  ->  -.  B  <  A
) )
193, 11, 183jaodan 1288 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  -.  B  <  A ) )
20 pnfnlt 9695 . . . . . . 7  |-  ( B  e.  RR*  ->  -. +oo  <  B )
2120adantl 275 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -. +oo  <  B )
22 breq1 3969 . . . . . . 7  |-  ( A  = +oo  ->  ( A  <  B  <-> +oo  <  B
) )
2322adantr 274 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  <-> +oo 
<  B ) )
2421, 23mtbird 663 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -.  A  <  B )
2524pm2.21d 609 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  ->  -.  B  <  A
) )
262, 25sylan2br 286 . . 3  |-  ( ( A  = +oo  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  -.  B  <  A ) )
27 rexr 7924 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
28 nltmnf 9696 . . . . . . . 8  |-  ( B  e.  RR*  ->  -.  B  < -oo )
2927, 28syl 14 . . . . . . 7  |-  ( B  e.  RR  ->  -.  B  < -oo )
3029adantl 275 . . . . . 6  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  -.  B  < -oo )
31 breq2 3970 . . . . . . 7  |-  ( A  = -oo  ->  ( B  <  A  <->  B  < -oo ) )
3231adantr 274 . . . . . 6  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( B  <  A  <->  B  < -oo ) )
3330, 32mtbird 663 . . . . 5  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  -.  B  <  A
)
3433a1d 22 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( A  <  B  ->  -.  B  <  A
) )
35 mnfxr 7935 . . . . . . . 8  |- -oo  e.  RR*
36 pnfnlt 9695 . . . . . . . 8  |-  ( -oo  e.  RR*  ->  -. +oo  < -oo )
3735, 36ax-mp 5 . . . . . . 7  |-  -. +oo  < -oo
38 breq12 3971 . . . . . . 7  |-  ( ( B  = +oo  /\  A  = -oo )  ->  ( B  <  A  <-> +oo 
< -oo ) )
3937, 38mtbiri 665 . . . . . 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 9687 . . . . . . 7  |-  ( -oo  e.  RR*  ->  -. -oo  < -oo )
4335, 42ax-mp 5 . . . . . 6  |-  -. -oo  < -oo
44 breq12 3971 . . . . . 6  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( A  <  B  <-> -oo 
< -oo ) )
4543, 44mtbiri 665 . . . . 5  |-  ( ( A  = -oo  /\  B  = -oo )  ->  -.  A  <  B
)
4645pm2.21d 609 . . . 4  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( A  <  B  ->  -.  B  <  A
) )
4734, 41, 463jaodan 1288 . . 3  |-  ( ( A  = -oo  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  -.  B  <  A ) )
4819, 26, 473jaoian 1287 . 2  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  -> 
( A  <  B  ->  -.  B  <  A
) )
491, 2, 48syl2anb 289 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 962    = wceq 1335    e. wcel 2128   class class class wbr 3966   RRcr 7732   +oocpnf 7910   -oocmnf 7911   RR*cxr 7912    < clt 7913
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-cnex 7824  ax-resscn 7825  ax-pre-ltirr 7845  ax-pre-lttrn 7847
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-xp 4593  df-pnf 7915  df-mnf 7916  df-xr 7917  df-ltxr 7918
This theorem is referenced by:  xrltnsym2  9702  xrltle  9706
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