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Theorem xrlttri 11334
Description: Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 9555 or axlttri 9645. (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 11319 . . . . . . . 8  |-  ( A  e.  RR*  ->  -.  A  <  A )
21adantr 465 . . . . . . 7  |-  ( ( A  e.  RR*  /\  A  =  B )  ->  -.  A  <  A )
3 breq2 4444 . . . . . . . 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 11332 . . . . 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 11314 . . . 4  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
12 elxr 11314 . . . 4  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
13 axlttri 9645 . . . . . . . 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 11320 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  < +oo )
1716adantr 465 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  A  < +oo )
18 breq2 4444 . . . . . . . . 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 11322 . . . . . . . . . 10  |-  ( A  e.  RR  -> -oo  <  A )
2322adantr 465 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = -oo )  -> -oo  <  A )
24 breq1 4443 . . . . . . . . . 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 1289 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( -.  A  <  B  -> 
( A  =  B  \/  B  <  A
) ) )
30 ltpnf 11320 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  < +oo )
3130adantl 466 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  e.  RR )  ->  B  < +oo )
32 breq2 4444 . . . . . . . . . 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 2488 . . . . . . . 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 11324 . . . . . . . . . 10  |- -oo  < +oo
41 breq12 4445 . . . . . . . . . 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 1289 . . . . 5  |-  ( ( A  = +oo  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( -.  A  <  B  -> 
( A  =  B  \/  B  <  A
) ) )
47 mnflt 11322 . . . . . . . . 9  |-  ( B  e.  RR  -> -oo  <  B )
4847adantl 466 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  e.  RR )  -> -oo  <  B )
49 breq1 4443 . . . . . . . . 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 4445 . . . . . . . 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 2488 . . . . . . . 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 1289 . . . . 5  |-  ( ( A  = -oo  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( -.  A  <  B  -> 
( A  =  B  \/  B  <  A
) ) )
6029, 46, 593jaoian 1288 . . . 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 967    = wceq 1374    e. wcel 1762   class class class wbr 4440   RRcr 9480   +oocpnf 9614   -oocmnf 9615   RR*cxr 9616    < clt 9617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-pre-lttri 9555  ax-pre-lttrn 9556
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622
This theorem is referenced by:  xrltso  11336  xrleloe  11339  xrltlen  11341
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