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Theorem xrlttri 10724
Description: Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 9056 or axlttri 9139. (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 10712 . . . . . . . 8  |-  ( A  e.  RR*  ->  -.  A  <  A )
21adantr 452 . . . . . . 7  |-  ( ( A  e.  RR*  /\  A  =  B )  ->  -.  A  <  A )
3 breq2 4208 . . . . . . . 8  |-  ( A  =  B  ->  ( A  <  A  <->  A  <  B ) )
43adantl 453 . . . . . . 7  |-  ( ( A  e.  RR*  /\  A  =  B )  ->  ( A  <  A  <->  A  <  B ) )
52, 4mtbid 292 . . . . . 6  |-  ( ( A  e.  RR*  /\  A  =  B )  ->  -.  A  <  B )
65ex 424 . . . . 5  |-  ( A  e.  RR*  ->  ( A  =  B  ->  -.  A  <  B ) )
76adantr 452 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  ->  -.  A  <  B ) )
8 xrltnsym 10722 . . . . 5  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  ( B  <  A  ->  -.  A  <  B ) )
98ancoms 440 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  <  A  ->  -.  A  <  B ) )
107, 9jaod 370 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  =  B  \/  B  <  A
)  ->  -.  A  <  B ) )
11 elxr 10708 . . . 4  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = 
+oo  \/  A  =  -oo ) )
12 elxr 10708 . . . 4  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = 
+oo  \/  B  =  -oo ) )
13 axlttri 9139 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  -.  ( A  =  B  \/  B  <  A
) ) )
1413biimprd 215 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  ( A  =  B  \/  B  <  A )  ->  A  <  B ) )
1514con1d 118 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
16 ltpnf 10713 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  <  +oo )
1716adantr 452 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  =  +oo )  ->  A  <  +oo )
18 breq2 4208 . . . . . . . . 9  |-  ( B  =  +oo  ->  ( A  <  B  <->  A  <  +oo ) )
1918adantl 453 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  =  +oo )  -> 
( A  <  B  <->  A  <  +oo ) )
2017, 19mpbird 224 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  =  +oo )  ->  A  <  B )
2120pm2.24d 137 . . . . . 6  |-  ( ( A  e.  RR  /\  B  =  +oo )  -> 
( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
22 mnflt 10714 . . . . . . . . . 10  |-  ( A  e.  RR  ->  -oo  <  A )
2322adantr 452 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  =  -oo )  ->  -oo  <  A )
24 breq1 4207 . . . . . . . . . 10  |-  ( B  =  -oo  ->  ( B  <  A  <->  -oo  <  A
) )
2524adantl 453 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  =  -oo )  -> 
( B  <  A  <->  -oo 
<  A ) )
2623, 25mpbird 224 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  =  -oo )  ->  B  <  A )
2726olcd 383 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  =  -oo )  -> 
( A  =  B  \/  B  <  A
) )
2827a1d 23 . . . . . 6  |-  ( ( A  e.  RR  /\  B  =  -oo )  -> 
( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
2915, 21, 283jaodan 1250 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo )
)  ->  ( -.  A  <  B  ->  ( A  =  B  \/  B  <  A ) ) )
30 ltpnf 10713 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  <  +oo )
3130adantl 453 . . . . . . . . 9  |-  ( ( A  =  +oo  /\  B  e.  RR )  ->  B  <  +oo )
32 breq2 4208 . . . . . . . . . 10  |-  ( A  =  +oo  ->  ( B  <  A  <->  B  <  +oo ) )
3332adantr 452 . . . . . . . . 9  |-  ( ( A  =  +oo  /\  B  e.  RR )  ->  ( B  <  A  <->  B  <  +oo ) )
3431, 33mpbird 224 . . . . . . . 8  |-  ( ( A  =  +oo  /\  B  e.  RR )  ->  B  <  A )
3534olcd 383 . . . . . . 7  |-  ( ( A  =  +oo  /\  B  e.  RR )  ->  ( A  =  B  \/  B  <  A
) )
3635a1d 23 . . . . . 6  |-  ( ( A  =  +oo  /\  B  e.  RR )  ->  ( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
37 eqtr3 2454 . . . . . . . 8  |-  ( ( A  =  +oo  /\  B  =  +oo )  ->  A  =  B )
3837orcd 382 . . . . . . 7  |-  ( ( A  =  +oo  /\  B  =  +oo )  -> 
( A  =  B  \/  B  <  A
) )
3938a1d 23 . . . . . 6  |-  ( ( A  =  +oo  /\  B  =  +oo )  -> 
( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
40 mnfltpnf 10715 . . . . . . . . . 10  |-  -oo  <  +oo
41 breq12 4209 . . . . . . . . . 10  |-  ( ( B  =  -oo  /\  A  =  +oo )  -> 
( B  <  A  <->  -oo 
<  +oo ) )
4240, 41mpbiri 225 . . . . . . . . 9  |-  ( ( B  =  -oo  /\  A  =  +oo )  ->  B  <  A )
4342ancoms 440 . . . . . . . 8  |-  ( ( A  =  +oo  /\  B  =  -oo )  ->  B  <  A )
4443olcd 383 . . . . . . 7  |-  ( ( A  =  +oo  /\  B  =  -oo )  -> 
( A  =  B  \/  B  <  A
) )
4544a1d 23 . . . . . 6  |-  ( ( A  =  +oo  /\  B  =  -oo )  -> 
( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
4636, 39, 453jaodan 1250 . . . . 5  |-  ( ( A  =  +oo  /\  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo )
)  ->  ( -.  A  <  B  ->  ( A  =  B  \/  B  <  A ) ) )
47 mnflt 10714 . . . . . . . . 9  |-  ( B  e.  RR  ->  -oo  <  B )
4847adantl 453 . . . . . . . 8  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  -oo  <  B )
49 breq1 4207 . . . . . . . . 9  |-  ( A  =  -oo  ->  ( A  <  B  <->  -oo  <  B
) )
5049adantr 452 . . . . . . . 8  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  ( A  <  B  <->  -oo 
<  B ) )
5148, 50mpbird 224 . . . . . . 7  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  A  <  B )
5251pm2.24d 137 . . . . . 6  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  ( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
53 breq12 4209 . . . . . . . 8  |-  ( ( A  =  -oo  /\  B  =  +oo )  -> 
( A  <  B  <->  -oo 
<  +oo ) )
5440, 53mpbiri 225 . . . . . . 7  |-  ( ( A  =  -oo  /\  B  =  +oo )  ->  A  <  B )
5554pm2.24d 137 . . . . . 6  |-  ( ( A  =  -oo  /\  B  =  +oo )  -> 
( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
56 eqtr3 2454 . . . . . . . 8  |-  ( ( A  =  -oo  /\  B  =  -oo )  ->  A  =  B )
5756orcd 382 . . . . . . 7  |-  ( ( A  =  -oo  /\  B  =  -oo )  -> 
( A  =  B  \/  B  <  A
) )
5857a1d 23 . . . . . 6  |-  ( ( A  =  -oo  /\  B  =  -oo )  -> 
( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
5952, 55, 583jaodan 1250 . . . . 5  |-  ( ( A  =  -oo  /\  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo )
)  ->  ( -.  A  <  B  ->  ( A  =  B  \/  B  <  A ) ) )
6029, 46, 593jaoian 1249 . . . 4  |-  ( ( ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo )  /\  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo )
)  ->  ( -.  A  <  B  ->  ( A  =  B  \/  B  <  A ) ) )
6111, 12, 60syl2anb 466 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  A  <  B  -> 
( A  =  B  \/  B  <  A
) ) )
6210, 61impbid 184 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  =  B  \/  B  <  A
)  <->  -.  A  <  B ) )
6362con2bid 320 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  -.  ( A  =  B  \/  B  <  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ 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:  xrltso  10726  xrleloe  10729  xrltlen  10731
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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