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Theorem xrltnsym 8714
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 8696 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 8696 . 2  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 ltnsym 7104 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  -.  B  <  A
) )
4 rexr 7071 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
5 pnfnlt 8708 . . . . . . . 8  |-  ( A  e.  RR*  ->  -. +oo  <  A )
64, 5syl 14 . . . . . . 7  |-  ( A  e.  RR  ->  -. +oo 
<  A )
76adantr 261 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  -. +oo  <  A
)
8 breq1 3767 . . . . . . 7  |-  ( B  = +oo  ->  ( B  <  A  <-> +oo  <  A
) )
98adantl 262 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( B  <  A  <-> +oo 
<  A ) )
107, 9mtbird 598 . . . . 5  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  -.  B  <  A
)
1110a1d 22 . . . 4  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A  <  B  ->  -.  B  <  A
) )
12 nltmnf 8709 . . . . . . . 8  |-  ( A  e.  RR*  ->  -.  A  < -oo )
134, 12syl 14 . . . . . . 7  |-  ( A  e.  RR  ->  -.  A  < -oo )
1413adantr 261 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  A  < -oo )
15 breq2 3768 . . . . . . 7  |-  ( B  = -oo  ->  ( A  <  B  <->  A  < -oo ) )
1615adantl 262 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  <->  A  < -oo ) )
1714, 16mtbird 598 . . . . 5  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  A  <  B
)
1817pm2.21d 549 . . . 4  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  ->  -.  B  <  A
) )
193, 11, 183jaodan 1201 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  -.  B  <  A ) )
20 pnfnlt 8708 . . . . . . 7  |-  ( B  e.  RR*  ->  -. +oo  <  B )
2120adantl 262 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -. +oo  <  B )
22 breq1 3767 . . . . . . 7  |-  ( A  = +oo  ->  ( A  <  B  <-> +oo  <  B
) )
2322adantr 261 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  <-> +oo 
<  B ) )
2421, 23mtbird 598 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -.  A  <  B )
2524pm2.21d 549 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  ->  -.  B  <  A
) )
262, 25sylan2br 272 . . 3  |-  ( ( A  = +oo  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  -.  B  <  A ) )
27 rexr 7071 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
28 nltmnf 8709 . . . . . . . 8  |-  ( B  e.  RR*  ->  -.  B  < -oo )
2927, 28syl 14 . . . . . . 7  |-  ( B  e.  RR  ->  -.  B  < -oo )
3029adantl 262 . . . . . 6  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  -.  B  < -oo )
31 breq2 3768 . . . . . . 7  |-  ( A  = -oo  ->  ( B  <  A  <->  B  < -oo ) )
3231adantr 261 . . . . . 6  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( B  <  A  <->  B  < -oo ) )
3330, 32mtbird 598 . . . . 5  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  -.  B  <  A
)
3433a1d 22 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( A  <  B  ->  -.  B  <  A
) )
35 mnfxr 8694 . . . . . . . 8  |- -oo  e.  RR*
36 pnfnlt 8708 . . . . . . . 8  |-  ( -oo  e.  RR*  ->  -. +oo  < -oo )
3735, 36ax-mp 7 . . . . . . 7  |-  -. +oo  < -oo
38 breq12 3769 . . . . . . 7  |-  ( ( B  = +oo  /\  A  = -oo )  ->  ( B  <  A  <-> +oo 
< -oo ) )
3937, 38mtbiri 600 . . . . . 6  |-  ( ( B  = +oo  /\  A  = -oo )  ->  -.  B  <  A
)
4039ancoms 255 . . . . 5  |-  ( ( A  = -oo  /\  B  = +oo )  ->  -.  B  <  A
)
4140a1d 22 . . . 4  |-  ( ( A  = -oo  /\  B  = +oo )  ->  ( A  <  B  ->  -.  B  <  A
) )
42 xrltnr 8701 . . . . . . 7  |-  ( -oo  e.  RR*  ->  -. -oo  < -oo )
4335, 42ax-mp 7 . . . . . 6  |-  -. -oo  < -oo
44 breq12 3769 . . . . . 6  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( A  <  B  <-> -oo 
< -oo ) )
4543, 44mtbiri 600 . . . . 5  |-  ( ( A  = -oo  /\  B  = -oo )  ->  -.  A  <  B
)
4645pm2.21d 549 . . . 4  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( A  <  B  ->  -.  B  <  A
) )
4734, 41, 463jaodan 1201 . . 3  |-  ( ( A  = -oo  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  -.  B  <  A ) )
4819, 26, 473jaoian 1200 . 2  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  -> 
( A  <  B  ->  -.  B  <  A
) )
491, 2, 48syl2anb 275 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  -.  B  <  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 97    <-> wb 98    \/ w3o 884    = wceq 1243    e. wcel 1393   class class class wbr 3764   RRcr 6888   +oocpnf 7057   -oocmnf 7058   RR*cxr 7059    < clt 7060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-pre-ltirr 6996  ax-pre-lttrn 6998
This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065
This theorem is referenced by:  xrltnsym2  8715  xrltle  8719
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