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Theorem xrltnsym 10722
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 10708 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = 
+oo  \/  A  =  -oo ) )
2 elxr 10708 . 2  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = 
+oo  \/  B  =  -oo ) )
3 ltnsym 9164 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  -.  B  <  A
) )
4 rexr 9122 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
5 pnfnlt 10717 . . . . . . . 8  |-  ( A  e.  RR*  ->  -.  +oo  <  A )
64, 5syl 16 . . . . . . 7  |-  ( A  e.  RR  ->  -.  +oo 
<  A )
76adantr 452 . . . . . 6  |-  ( ( A  e.  RR  /\  B  =  +oo )  ->  -.  +oo  <  A )
8 breq1 4207 . . . . . . 7  |-  ( B  =  +oo  ->  ( B  <  A  <->  +oo  <  A
) )
98adantl 453 . . . . . 6  |-  ( ( A  e.  RR  /\  B  =  +oo )  -> 
( B  <  A  <->  +oo 
<  A ) )
107, 9mtbird 293 . . . . 5  |-  ( ( A  e.  RR  /\  B  =  +oo )  ->  -.  B  <  A )
1110a1d 23 . . . 4  |-  ( ( A  e.  RR  /\  B  =  +oo )  -> 
( A  <  B  ->  -.  B  <  A
) )
12 nltmnf 10718 . . . . . . . 8  |-  ( A  e.  RR*  ->  -.  A  <  -oo )
134, 12syl 16 . . . . . . 7  |-  ( A  e.  RR  ->  -.  A  <  -oo )
1413adantr 452 . . . . . 6  |-  ( ( A  e.  RR  /\  B  =  -oo )  ->  -.  A  <  -oo )
15 breq2 4208 . . . . . . 7  |-  ( B  =  -oo  ->  ( A  <  B  <->  A  <  -oo ) )
1615adantl 453 . . . . . 6  |-  ( ( A  e.  RR  /\  B  =  -oo )  -> 
( A  <  B  <->  A  <  -oo ) )
1714, 16mtbird 293 . . . . 5  |-  ( ( A  e.  RR  /\  B  =  -oo )  ->  -.  A  <  B )
1817pm2.21d 100 . . . 4  |-  ( ( A  e.  RR  /\  B  =  -oo )  -> 
( A  <  B  ->  -.  B  <  A
) )
193, 11, 183jaodan 1250 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo )
)  ->  ( A  <  B  ->  -.  B  <  A ) )
20 pnfnlt 10717 . . . . . . 7  |-  ( B  e.  RR*  ->  -.  +oo  <  B )
2120adantl 453 . . . . . 6  |-  ( ( A  =  +oo  /\  B  e.  RR* )  ->  -.  +oo  <  B )
22 breq1 4207 . . . . . . 7  |-  ( A  =  +oo  ->  ( A  <  B  <->  +oo  <  B
) )
2322adantr 452 . . . . . 6  |-  ( ( A  =  +oo  /\  B  e.  RR* )  -> 
( A  <  B  <->  +oo 
<  B ) )
2421, 23mtbird 293 . . . . 5  |-  ( ( A  =  +oo  /\  B  e.  RR* )  ->  -.  A  <  B )
2524pm2.21d 100 . . . 4  |-  ( ( A  =  +oo  /\  B  e.  RR* )  -> 
( A  <  B  ->  -.  B  <  A
) )
262, 25sylan2br 463 . . 3  |-  ( ( A  =  +oo  /\  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo )
)  ->  ( A  <  B  ->  -.  B  <  A ) )
27 rexr 9122 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
28 nltmnf 10718 . . . . . . . 8  |-  ( B  e.  RR*  ->  -.  B  <  -oo )
2927, 28syl 16 . . . . . . 7  |-  ( B  e.  RR  ->  -.  B  <  -oo )
3029adantl 453 . . . . . 6  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  -.  B  <  -oo )
31 breq2 4208 . . . . . . 7  |-  ( A  =  -oo  ->  ( B  <  A  <->  B  <  -oo ) )
3231adantr 452 . . . . . 6  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  ( B  <  A  <->  B  <  -oo ) )
3330, 32mtbird 293 . . . . 5  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  -.  B  <  A
)
3433a1d 23 . . . 4  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  ( A  <  B  ->  -.  B  <  A
) )
35 mnfxr 10706 . . . . . . . 8  |-  -oo  e.  RR*
36 pnfnlt 10717 . . . . . . . 8  |-  (  -oo  e.  RR*  ->  -.  +oo  <  -oo )
3735, 36ax-mp 8 . . . . . . 7  |-  -.  +oo  <  -oo
38 breq12 4209 . . . . . . 7  |-  ( ( B  =  +oo  /\  A  =  -oo )  -> 
( B  <  A  <->  +oo 
<  -oo ) )
3937, 38mtbiri 295 . . . . . 6  |-  ( ( B  =  +oo  /\  A  =  -oo )  ->  -.  B  <  A )
4039ancoms 440 . . . . 5  |-  ( ( A  =  -oo  /\  B  =  +oo )  ->  -.  B  <  A )
4140a1d 23 . . . 4  |-  ( ( A  =  -oo  /\  B  =  +oo )  -> 
( A  <  B  ->  -.  B  <  A
) )
42 xrltnr 10712 . . . . . . 7  |-  (  -oo  e.  RR*  ->  -.  -oo  <  -oo )
4335, 42ax-mp 8 . . . . . 6  |-  -.  -oo  <  -oo
44 breq12 4209 . . . . . 6  |-  ( ( A  =  -oo  /\  B  =  -oo )  -> 
( A  <  B  <->  -oo 
<  -oo ) )
4543, 44mtbiri 295 . . . . 5  |-  ( ( A  =  -oo  /\  B  =  -oo )  ->  -.  A  <  B )
4645pm2.21d 100 . . . 4  |-  ( ( A  =  -oo  /\  B  =  -oo )  -> 
( A  <  B  ->  -.  B  <  A
) )
4734, 41, 463jaodan 1250 . . 3  |-  ( ( A  =  -oo  /\  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo )
)  ->  ( A  <  B  ->  -.  B  <  A ) )
4819, 26, 473jaoian 1249 . 2  |-  ( ( ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo )  /\  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo )
)  ->  ( A  <  B  ->  -.  B  <  A ) )
491, 2, 48syl2anb 466 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  -.  B  <  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    \/ w3o 935    = wceq 1652    e. wcel 1725   class class class wbr 4204   RRcr 8981    +oocpnf 9109    -oocmnf 9110   RR*cxr 9111    < clt 9112
This theorem is referenced by:  xrltnsym2  10723  xrlttri  10724  xmullem2  10836  sgnp  28457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-pre-lttri 9056  ax-pre-lttrn 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117
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