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Theorem xrltnsym 10735
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 10721 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = 
+oo  \/  A  =  -oo ) )
2 elxr 10721 . 2  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = 
+oo  \/  B  =  -oo ) )
3 ltnsym 9177 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  -.  B  <  A
) )
4 rexr 9135 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
5 pnfnlt 10730 . . . . . . . 8  |-  ( A  e.  RR*  ->  -.  +oo  <  A )
64, 5syl 16 . . . . . . 7  |-  ( A  e.  RR  ->  -.  +oo 
<  A )
76adantr 453 . . . . . 6  |-  ( ( A  e.  RR  /\  B  =  +oo )  ->  -.  +oo  <  A )
8 breq1 4218 . . . . . . 7  |-  ( B  =  +oo  ->  ( B  <  A  <->  +oo  <  A
) )
98adantl 454 . . . . . 6  |-  ( ( A  e.  RR  /\  B  =  +oo )  -> 
( B  <  A  <->  +oo 
<  A ) )
107, 9mtbird 294 . . . . 5  |-  ( ( A  e.  RR  /\  B  =  +oo )  ->  -.  B  <  A )
1110a1d 24 . . . 4  |-  ( ( A  e.  RR  /\  B  =  +oo )  -> 
( A  <  B  ->  -.  B  <  A
) )
12 nltmnf 10731 . . . . . . . 8  |-  ( A  e.  RR*  ->  -.  A  <  -oo )
134, 12syl 16 . . . . . . 7  |-  ( A  e.  RR  ->  -.  A  <  -oo )
1413adantr 453 . . . . . 6  |-  ( ( A  e.  RR  /\  B  =  -oo )  ->  -.  A  <  -oo )
15 breq2 4219 . . . . . . 7  |-  ( B  =  -oo  ->  ( A  <  B  <->  A  <  -oo ) )
1615adantl 454 . . . . . 6  |-  ( ( A  e.  RR  /\  B  =  -oo )  -> 
( A  <  B  <->  A  <  -oo ) )
1714, 16mtbird 294 . . . . 5  |-  ( ( A  e.  RR  /\  B  =  -oo )  ->  -.  A  <  B )
1817pm2.21d 101 . . . 4  |-  ( ( A  e.  RR  /\  B  =  -oo )  -> 
( A  <  B  ->  -.  B  <  A
) )
193, 11, 183jaodan 1251 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo )
)  ->  ( A  <  B  ->  -.  B  <  A ) )
20 pnfnlt 10730 . . . . . . 7  |-  ( B  e.  RR*  ->  -.  +oo  <  B )
2120adantl 454 . . . . . 6  |-  ( ( A  =  +oo  /\  B  e.  RR* )  ->  -.  +oo  <  B )
22 breq1 4218 . . . . . . 7  |-  ( A  =  +oo  ->  ( A  <  B  <->  +oo  <  B
) )
2322adantr 453 . . . . . 6  |-  ( ( A  =  +oo  /\  B  e.  RR* )  -> 
( A  <  B  <->  +oo 
<  B ) )
2421, 23mtbird 294 . . . . 5  |-  ( ( A  =  +oo  /\  B  e.  RR* )  ->  -.  A  <  B )
2524pm2.21d 101 . . . 4  |-  ( ( A  =  +oo  /\  B  e.  RR* )  -> 
( A  <  B  ->  -.  B  <  A
) )
262, 25sylan2br 464 . . 3  |-  ( ( A  =  +oo  /\  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo )
)  ->  ( A  <  B  ->  -.  B  <  A ) )
27 rexr 9135 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
28 nltmnf 10731 . . . . . . . 8  |-  ( B  e.  RR*  ->  -.  B  <  -oo )
2927, 28syl 16 . . . . . . 7  |-  ( B  e.  RR  ->  -.  B  <  -oo )
3029adantl 454 . . . . . 6  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  -.  B  <  -oo )
31 breq2 4219 . . . . . . 7  |-  ( A  =  -oo  ->  ( B  <  A  <->  B  <  -oo ) )
3231adantr 453 . . . . . 6  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  ( B  <  A  <->  B  <  -oo ) )
3330, 32mtbird 294 . . . . 5  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  -.  B  <  A
)
3433a1d 24 . . . 4  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  ( A  <  B  ->  -.  B  <  A
) )
35 mnfxr 10719 . . . . . . . 8  |-  -oo  e.  RR*
36 pnfnlt 10730 . . . . . . . 8  |-  (  -oo  e.  RR*  ->  -.  +oo  <  -oo )
3735, 36ax-mp 5 . . . . . . 7  |-  -.  +oo  <  -oo
38 breq12 4220 . . . . . . 7  |-  ( ( B  =  +oo  /\  A  =  -oo )  -> 
( B  <  A  <->  +oo 
<  -oo ) )
3937, 38mtbiri 296 . . . . . 6  |-  ( ( B  =  +oo  /\  A  =  -oo )  ->  -.  B  <  A )
4039ancoms 441 . . . . 5  |-  ( ( A  =  -oo  /\  B  =  +oo )  ->  -.  B  <  A )
4140a1d 24 . . . 4  |-  ( ( A  =  -oo  /\  B  =  +oo )  -> 
( A  <  B  ->  -.  B  <  A
) )
42 xrltnr 10725 . . . . . . 7  |-  (  -oo  e.  RR*  ->  -.  -oo  <  -oo )
4335, 42ax-mp 5 . . . . . 6  |-  -.  -oo  <  -oo
44 breq12 4220 . . . . . 6  |-  ( ( A  =  -oo  /\  B  =  -oo )  -> 
( A  <  B  <->  -oo 
<  -oo ) )
4543, 44mtbiri 296 . . . . 5  |-  ( ( A  =  -oo  /\  B  =  -oo )  ->  -.  A  <  B )
4645pm2.21d 101 . . . 4  |-  ( ( A  =  -oo  /\  B  =  -oo )  -> 
( A  <  B  ->  -.  B  <  A
) )
4734, 41, 463jaodan 1251 . . 3  |-  ( ( A  =  -oo  /\  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo )
)  ->  ( A  <  B  ->  -.  B  <  A ) )
4819, 26, 473jaoian 1250 . 2  |-  ( ( ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo )  /\  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo )
)  ->  ( A  <  B  ->  -.  B  <  A ) )
491, 2, 48syl2anb 467 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  -.  B  <  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    \/ w3o 936    = wceq 1653    e. wcel 1726   class class class wbr 4215   RRcr 8994    +oocpnf 9122    -oocmnf 9123   RR*cxr 9124    < clt 9125
This theorem is referenced by:  xrltnsym2  10736  xrlttri  10737  xmullem2  10849  sgnp  28594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-pre-lttri 9069  ax-pre-lttrn 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130
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