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Theorem xrlttri 11266
Description: Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 9477 or axlttri 9567. (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 11251 . . . . . . . 8  |-  ( A  e.  RR*  ->  -.  A  <  A )
21adantr 463 . . . . . . 7  |-  ( ( A  e.  RR*  /\  A  =  B )  ->  -.  A  <  A )
3 breq2 4371 . . . . . . . 8  |-  ( A  =  B  ->  ( A  <  A  <->  A  <  B ) )
43adantl 464 . . . . . . 7  |-  ( ( A  e.  RR*  /\  A  =  B )  ->  ( A  <  A  <->  A  <  B ) )
52, 4mtbid 298 . . . . . 6  |-  ( ( A  e.  RR*  /\  A  =  B )  ->  -.  A  <  B )
65ex 432 . . . . 5  |-  ( A  e.  RR*  ->  ( A  =  B  ->  -.  A  <  B ) )
76adantr 463 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  ->  -.  A  <  B ) )
8 xrltnsym 11264 . . . . 5  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  ( B  <  A  ->  -.  A  <  B ) )
98ancoms 451 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  <  A  ->  -.  A  <  B ) )
107, 9jaod 378 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  =  B  \/  B  <  A
)  ->  -.  A  <  B ) )
11 elxr 11246 . . . 4  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
12 elxr 11246 . . . 4  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
13 axlttri 9567 . . . . . . . 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 11252 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  < +oo )
1716adantr 463 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  A  < +oo )
18 breq2 4371 . . . . . . . . 9  |-  ( B  = +oo  ->  ( A  <  B  <->  A  < +oo ) )
1918adantl 464 . . . . . . . 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 11254 . . . . . . . . . 10  |-  ( A  e.  RR  -> -oo  <  A )
2322adantr 463 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = -oo )  -> -oo  <  A )
24 breq1 4370 . . . . . . . . . 10  |-  ( B  = -oo  ->  ( B  <  A  <-> -oo  <  A
) )
2524adantl 464 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( B  <  A  <-> -oo 
<  A ) )
2623, 25mpbird 232 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  B  <  A )
2726olcd 391 . . . . . . 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 1292 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( -.  A  <  B  -> 
( A  =  B  \/  B  <  A
) ) )
30 ltpnf 11252 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  < +oo )
3130adantl 464 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  e.  RR )  ->  B  < +oo )
32 breq2 4371 . . . . . . . . . 10  |-  ( A  = +oo  ->  ( B  <  A  <->  B  < +oo ) )
3332adantr 463 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  e.  RR )  ->  ( B  <  A  <->  B  < +oo ) )
3431, 33mpbird 232 . . . . . . . 8  |-  ( ( A  = +oo  /\  B  e.  RR )  ->  B  <  A )
3534olcd 391 . . . . . . 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 2410 . . . . . . . 8  |-  ( ( A  = +oo  /\  B  = +oo )  ->  A  =  B )
3837orcd 390 . . . . . . 7  |-  ( ( A  = +oo  /\  B  = +oo )  ->  ( A  =  B  \/  B  <  A
) )
3938a1d 25 . . . . . 6  |-  ( ( A  = +oo  /\  B  = +oo )  ->  ( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
40 mnfltpnf 11256 . . . . . . . . . 10  |- -oo  < +oo
41 breq12 4372 . . . . . . . . . 10  |-  ( ( B  = -oo  /\  A  = +oo )  ->  ( B  <  A  <-> -oo 
< +oo ) )
4240, 41mpbiri 233 . . . . . . . . 9  |-  ( ( B  = -oo  /\  A  = +oo )  ->  B  <  A )
4342ancoms 451 . . . . . . . 8  |-  ( ( A  = +oo  /\  B  = -oo )  ->  B  <  A )
4443olcd 391 . . . . . . 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 1292 . . . . 5  |-  ( ( A  = +oo  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( -.  A  <  B  -> 
( A  =  B  \/  B  <  A
) ) )
47 mnflt 11254 . . . . . . . . 9  |-  ( B  e.  RR  -> -oo  <  B )
4847adantl 464 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  e.  RR )  -> -oo  <  B )
49 breq1 4370 . . . . . . . . 9  |-  ( A  = -oo  ->  ( A  <  B  <-> -oo  <  B
) )
5049adantr 463 . . . . . . . 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 4372 . . . . . . . 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 2410 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  = -oo )  ->  A  =  B )
5756orcd 390 . . . . . . 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 1292 . . . . 5  |-  ( ( A  = -oo  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( -.  A  <  B  -> 
( A  =  B  \/  B  <  A
) ) )
6029, 46, 593jaoian 1291 . . . 4  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  -> 
( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
6111, 12, 60syl2anb 477 . . 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 327 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 366    /\ wa 367    \/ w3o 970    = wceq 1399    e. wcel 1826   class class class wbr 4367   RRcr 9402   +oocpnf 9536   -oocmnf 9537   RR*cxr 9538    < clt 9539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-pre-lttri 9477  ax-pre-lttrn 9478
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-po 4714  df-so 4715  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544
This theorem is referenced by:  xrltso  11268  xrleloe  11271  xrltlen  11273
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