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Theorem xrltnsym 9750
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 9733 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 9733 . 2  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 ltnsym 8005 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  -.  B  <  A
) )
4 rexr 7965 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
5 pnfnlt 9744 . . . . . . . 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 3992 . . . . . . 7  |-  ( B  = +oo  ->  ( B  <  A  <-> +oo  <  A
) )
98adantl 275 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( B  <  A  <-> +oo 
<  A ) )
107, 9mtbird 668 . . . . 5  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  -.  B  <  A
)
1110a1d 22 . . . 4  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A  <  B  ->  -.  B  <  A
) )
12 nltmnf 9745 . . . . . . . 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 3993 . . . . . . 7  |-  ( B  = -oo  ->  ( A  <  B  <->  A  < -oo ) )
1615adantl 275 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  <->  A  < -oo ) )
1714, 16mtbird 668 . . . . 5  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  A  <  B
)
1817pm2.21d 614 . . . 4  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  ->  -.  B  <  A
) )
193, 11, 183jaodan 1301 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  -.  B  <  A ) )
20 pnfnlt 9744 . . . . . . 7  |-  ( B  e.  RR*  ->  -. +oo  <  B )
2120adantl 275 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -. +oo  <  B )
22 breq1 3992 . . . . . . 7  |-  ( A  = +oo  ->  ( A  <  B  <-> +oo  <  B
) )
2322adantr 274 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  <-> +oo 
<  B ) )
2421, 23mtbird 668 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -.  A  <  B )
2524pm2.21d 614 . . . 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 7965 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
28 nltmnf 9745 . . . . . . . 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 3993 . . . . . . 7  |-  ( A  = -oo  ->  ( B  <  A  <->  B  < -oo ) )
3231adantr 274 . . . . . 6  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( B  <  A  <->  B  < -oo ) )
3330, 32mtbird 668 . . . . 5  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  -.  B  <  A
)
3433a1d 22 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( A  <  B  ->  -.  B  <  A
) )
35 mnfxr 7976 . . . . . . . 8  |- -oo  e.  RR*
36 pnfnlt 9744 . . . . . . . 8  |-  ( -oo  e.  RR*  ->  -. +oo  < -oo )
3735, 36ax-mp 5 . . . . . . 7  |-  -. +oo  < -oo
38 breq12 3994 . . . . . . 7  |-  ( ( B  = +oo  /\  A  = -oo )  ->  ( B  <  A  <-> +oo 
< -oo ) )
3937, 38mtbiri 670 . . . . . 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 9736 . . . . . . 7  |-  ( -oo  e.  RR*  ->  -. -oo  < -oo )
4335, 42ax-mp 5 . . . . . 6  |-  -. -oo  < -oo
44 breq12 3994 . . . . . 6  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( A  <  B  <-> -oo 
< -oo ) )
4543, 44mtbiri 670 . . . . 5  |-  ( ( A  = -oo  /\  B  = -oo )  ->  -.  A  <  B
)
4645pm2.21d 614 . . . 4  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( A  <  B  ->  -.  B  <  A
) )
4734, 41, 463jaodan 1301 . . 3  |-  ( ( A  = -oo  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  -.  B  <  A ) )
4819, 26, 473jaoian 1300 . 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 972    = wceq 1348    e. wcel 2141   class class class wbr 3989   RRcr 7773   +oocpnf 7951   -oocmnf 7952   RR*cxr 7953    < clt 7954
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-pre-ltirr 7886  ax-pre-lttrn 7888
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959
This theorem is referenced by:  xrltnsym2  9751  xrltle  9755
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