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Theorem xrlttri 11108
Description: Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 9348 or axlttri 9438. (Contributed by NM, 14-Oct-2005.)
Assertion
Ref Expression
xrlttri  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  -.  ( A  =  B  \/  B  <  A ) ) )

Proof of Theorem xrlttri
StepHypRef Expression
1 xrltnr 11093 . . . . . . . 8  |-  ( A  e.  RR*  ->  -.  A  <  A )
21adantr 465 . . . . . . 7  |-  ( ( A  e.  RR*  /\  A  =  B )  ->  -.  A  <  A )
3 breq2 4291 . . . . . . . 8  |-  ( A  =  B  ->  ( A  <  A  <->  A  <  B ) )
43adantl 466 . . . . . . 7  |-  ( ( A  e.  RR*  /\  A  =  B )  ->  ( A  <  A  <->  A  <  B ) )
52, 4mtbid 300 . . . . . 6  |-  ( ( A  e.  RR*  /\  A  =  B )  ->  -.  A  <  B )
65ex 434 . . . . 5  |-  ( A  e.  RR*  ->  ( A  =  B  ->  -.  A  <  B ) )
76adantr 465 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  ->  -.  A  <  B ) )
8 xrltnsym 11106 . . . . 5  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  ( B  <  A  ->  -.  A  <  B ) )
98ancoms 453 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  <  A  ->  -.  A  <  B ) )
107, 9jaod 380 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  =  B  \/  B  <  A
)  ->  -.  A  <  B ) )
11 elxr 11088 . . . 4  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
12 elxr 11088 . . . 4  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
13 axlttri 9438 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  -.  ( A  =  B  \/  B  <  A
) ) )
1413biimprd 223 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  ( A  =  B  \/  B  <  A )  ->  A  <  B ) )
1514con1d 124 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
16 ltpnf 11094 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  < +oo )
1716adantr 465 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  A  < +oo )
18 breq2 4291 . . . . . . . . 9  |-  ( B  = +oo  ->  ( A  <  B  <->  A  < +oo ) )
1918adantl 466 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A  <  B  <->  A  < +oo ) )
2017, 19mpbird 232 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  A  <  B )
2120pm2.24d 143 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
22 mnflt 11096 . . . . . . . . . 10  |-  ( A  e.  RR  -> -oo  <  A )
2322adantr 465 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = -oo )  -> -oo  <  A )
24 breq1 4290 . . . . . . . . . 10  |-  ( B  = -oo  ->  ( B  <  A  <-> -oo  <  A
) )
2524adantl 466 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( B  <  A  <-> -oo 
<  A ) )
2623, 25mpbird 232 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  B  <  A )
2726olcd 393 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  =  B  \/  B  <  A
) )
2827a1d 25 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
2915, 21, 283jaodan 1284 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( -.  A  <  B  -> 
( A  =  B  \/  B  <  A
) ) )
30 ltpnf 11094 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  < +oo )
3130adantl 466 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  e.  RR )  ->  B  < +oo )
32 breq2 4291 . . . . . . . . . 10  |-  ( A  = +oo  ->  ( B  <  A  <->  B  < +oo ) )
3332adantr 465 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  e.  RR )  ->  ( B  <  A  <->  B  < +oo ) )
3431, 33mpbird 232 . . . . . . . 8  |-  ( ( A  = +oo  /\  B  e.  RR )  ->  B  <  A )
3534olcd 393 . . . . . . 7  |-  ( ( A  = +oo  /\  B  e.  RR )  ->  ( A  =  B  \/  B  <  A
) )
3635a1d 25 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR )  ->  ( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
37 eqtr3 2457 . . . . . . . 8  |-  ( ( A  = +oo  /\  B  = +oo )  ->  A  =  B )
3837orcd 392 . . . . . . 7  |-  ( ( A  = +oo  /\  B  = +oo )  ->  ( A  =  B  \/  B  <  A
) )
3938a1d 25 . . . . . 6  |-  ( ( A  = +oo  /\  B  = +oo )  ->  ( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
40 mnfltpnf 11098 . . . . . . . . . 10  |- -oo  < +oo
41 breq12 4292 . . . . . . . . . 10  |-  ( ( B  = -oo  /\  A  = +oo )  ->  ( B  <  A  <-> -oo 
< +oo ) )
4240, 41mpbiri 233 . . . . . . . . 9  |-  ( ( B  = -oo  /\  A  = +oo )  ->  B  <  A )
4342ancoms 453 . . . . . . . 8  |-  ( ( A  = +oo  /\  B  = -oo )  ->  B  <  A )
4443olcd 393 . . . . . . 7  |-  ( ( A  = +oo  /\  B  = -oo )  ->  ( A  =  B  \/  B  <  A
) )
4544a1d 25 . . . . . 6  |-  ( ( A  = +oo  /\  B  = -oo )  ->  ( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
4636, 39, 453jaodan 1284 . . . . 5  |-  ( ( A  = +oo  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( -.  A  <  B  -> 
( A  =  B  \/  B  <  A
) ) )
47 mnflt 11096 . . . . . . . . 9  |-  ( B  e.  RR  -> -oo  <  B )
4847adantl 466 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  e.  RR )  -> -oo  <  B )
49 breq1 4290 . . . . . . . . 9  |-  ( A  = -oo  ->  ( A  <  B  <-> -oo  <  B
) )
5049adantr 465 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( A  <  B  <-> -oo 
<  B ) )
5148, 50mpbird 232 . . . . . . 7  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  A  <  B )
5251pm2.24d 143 . . . . . 6  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
53 breq12 4292 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  = +oo )  ->  ( A  <  B  <-> -oo 
< +oo ) )
5440, 53mpbiri 233 . . . . . . 7  |-  ( ( A  = -oo  /\  B  = +oo )  ->  A  <  B )
5554pm2.24d 143 . . . . . 6  |-  ( ( A  = -oo  /\  B  = +oo )  ->  ( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
56 eqtr3 2457 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  = -oo )  ->  A  =  B )
5756orcd 392 . . . . . . 7  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( A  =  B  \/  B  <  A
) )
5857a1d 25 . . . . . 6  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
5952, 55, 583jaodan 1284 . . . . 5  |-  ( ( A  = -oo  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( -.  A  <  B  -> 
( A  =  B  \/  B  <  A
) ) )
6029, 46, 593jaoian 1283 . . . 4  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  -> 
( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
6111, 12, 60syl2anb 479 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  A  <  B  -> 
( A  =  B  \/  B  <  A
) ) )
6210, 61impbid 191 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  =  B  \/  B  <  A
)  <->  -.  A  <  B ) )
6362con2bid 329 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  -.  ( A  =  B  \/  B  <  A ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 964    = wceq 1369    e. wcel 1756   class class class wbr 4287   RRcr 9273   +oocpnf 9407   -oocmnf 9408   RR*cxr 9409    < clt 9410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-pre-lttri 9348  ax-pre-lttrn 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415
This theorem is referenced by:  xrltso  11110  xrleloe  11113  xrltlen  11115
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