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Theorem xrlttri 10475
Description: Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 8813 or axlttri 8896. (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 10464 . . . . . . . 8  |-  ( A  e.  RR*  ->  -.  A  <  A )
21adantr 451 . . . . . . 7  |-  ( ( A  e.  RR*  /\  A  =  B )  ->  -.  A  <  A )
3 breq2 4029 . . . . . . . 8  |-  ( A  =  B  ->  ( A  <  A  <->  A  <  B ) )
43adantl 452 . . . . . . 7  |-  ( ( A  e.  RR*  /\  A  =  B )  ->  ( A  <  A  <->  A  <  B ) )
52, 4mtbid 291 . . . . . 6  |-  ( ( A  e.  RR*  /\  A  =  B )  ->  -.  A  <  B )
65ex 423 . . . . 5  |-  ( A  e.  RR*  ->  ( A  =  B  ->  -.  A  <  B ) )
76adantr 451 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  ->  -.  A  <  B ) )
8 xrltnsym 10473 . . . . 5  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  ( B  <  A  ->  -.  A  <  B ) )
98ancoms 439 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  <  A  ->  -.  A  <  B ) )
107, 9jaod 369 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  =  B  \/  B  <  A
)  ->  -.  A  <  B ) )
11 elxr 10460 . . . 4  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = 
+oo  \/  A  =  -oo ) )
12 elxr 10460 . . . 4  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = 
+oo  \/  B  =  -oo ) )
13 axlttri 8896 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  -.  ( A  =  B  \/  B  <  A
) ) )
1413biimprd 214 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  ( A  =  B  \/  B  <  A )  ->  A  <  B ) )
1514con1d 116 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
16 ltpnf 10465 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  <  +oo )
1716adantr 451 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  =  +oo )  ->  A  <  +oo )
18 breq2 4029 . . . . . . . . 9  |-  ( B  =  +oo  ->  ( A  <  B  <->  A  <  +oo ) )
1918adantl 452 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  =  +oo )  -> 
( A  <  B  <->  A  <  +oo ) )
2017, 19mpbird 223 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  =  +oo )  ->  A  <  B )
2120pm2.24d 135 . . . . . 6  |-  ( ( A  e.  RR  /\  B  =  +oo )  -> 
( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
22 mnflt 10466 . . . . . . . . . 10  |-  ( A  e.  RR  ->  -oo  <  A )
2322adantr 451 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  =  -oo )  ->  -oo  <  A )
24 breq1 4028 . . . . . . . . . 10  |-  ( B  =  -oo  ->  ( B  <  A  <->  -oo  <  A
) )
2524adantl 452 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  =  -oo )  -> 
( B  <  A  <->  -oo 
<  A ) )
2623, 25mpbird 223 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  =  -oo )  ->  B  <  A )
2726olcd 382 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  =  -oo )  -> 
( A  =  B  \/  B  <  A
) )
2827a1d 22 . . . . . 6  |-  ( ( A  e.  RR  /\  B  =  -oo )  -> 
( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
2915, 21, 283jaodan 1248 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo )
)  ->  ( -.  A  <  B  ->  ( A  =  B  \/  B  <  A ) ) )
30 ltpnf 10465 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  <  +oo )
3130adantl 452 . . . . . . . . 9  |-  ( ( A  =  +oo  /\  B  e.  RR )  ->  B  <  +oo )
32 breq2 4029 . . . . . . . . . 10  |-  ( A  =  +oo  ->  ( B  <  A  <->  B  <  +oo ) )
3332adantr 451 . . . . . . . . 9  |-  ( ( A  =  +oo  /\  B  e.  RR )  ->  ( B  <  A  <->  B  <  +oo ) )
3431, 33mpbird 223 . . . . . . . 8  |-  ( ( A  =  +oo  /\  B  e.  RR )  ->  B  <  A )
3534olcd 382 . . . . . . 7  |-  ( ( A  =  +oo  /\  B  e.  RR )  ->  ( A  =  B  \/  B  <  A
) )
3635a1d 22 . . . . . 6  |-  ( ( A  =  +oo  /\  B  e.  RR )  ->  ( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
37 eqtr3 2304 . . . . . . . 8  |-  ( ( A  =  +oo  /\  B  =  +oo )  ->  A  =  B )
3837orcd 381 . . . . . . 7  |-  ( ( A  =  +oo  /\  B  =  +oo )  -> 
( A  =  B  \/  B  <  A
) )
3938a1d 22 . . . . . 6  |-  ( ( A  =  +oo  /\  B  =  +oo )  -> 
( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
40 mnfltpnf 10467 . . . . . . . . . 10  |-  -oo  <  +oo
41 breq12 4030 . . . . . . . . . 10  |-  ( ( B  =  -oo  /\  A  =  +oo )  -> 
( B  <  A  <->  -oo 
<  +oo ) )
4240, 41mpbiri 224 . . . . . . . . 9  |-  ( ( B  =  -oo  /\  A  =  +oo )  ->  B  <  A )
4342ancoms 439 . . . . . . . 8  |-  ( ( A  =  +oo  /\  B  =  -oo )  ->  B  <  A )
4443olcd 382 . . . . . . 7  |-  ( ( A  =  +oo  /\  B  =  -oo )  -> 
( A  =  B  \/  B  <  A
) )
4544a1d 22 . . . . . 6  |-  ( ( A  =  +oo  /\  B  =  -oo )  -> 
( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
4636, 39, 453jaodan 1248 . . . . 5  |-  ( ( A  =  +oo  /\  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo )
)  ->  ( -.  A  <  B  ->  ( A  =  B  \/  B  <  A ) ) )
47 mnflt 10466 . . . . . . . . 9  |-  ( B  e.  RR  ->  -oo  <  B )
4847adantl 452 . . . . . . . 8  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  -oo  <  B )
49 breq1 4028 . . . . . . . . 9  |-  ( A  =  -oo  ->  ( A  <  B  <->  -oo  <  B
) )
5049adantr 451 . . . . . . . 8  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  ( A  <  B  <->  -oo 
<  B ) )
5148, 50mpbird 223 . . . . . . 7  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  A  <  B )
5251pm2.24d 135 . . . . . 6  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  ( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
53 breq12 4030 . . . . . . . 8  |-  ( ( A  =  -oo  /\  B  =  +oo )  -> 
( A  <  B  <->  -oo 
<  +oo ) )
5440, 53mpbiri 224 . . . . . . 7  |-  ( ( A  =  -oo  /\  B  =  +oo )  ->  A  <  B )
5554pm2.24d 135 . . . . . 6  |-  ( ( A  =  -oo  /\  B  =  +oo )  -> 
( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
56 eqtr3 2304 . . . . . . . 8  |-  ( ( A  =  -oo  /\  B  =  -oo )  ->  A  =  B )
5756orcd 381 . . . . . . 7  |-  ( ( A  =  -oo  /\  B  =  -oo )  -> 
( A  =  B  \/  B  <  A
) )
5857a1d 22 . . . . . 6  |-  ( ( A  =  -oo  /\  B  =  -oo )  -> 
( -.  A  < 
B  ->  ( A  =  B  \/  B  <  A ) ) )
5952, 55, 583jaodan 1248 . . . . 5  |-  ( ( A  =  -oo  /\  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo )
)  ->  ( -.  A  <  B  ->  ( A  =  B  \/  B  <  A ) ) )
6029, 46, 593jaoian 1247 . . . 4  |-  ( ( ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo )  /\  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo )
)  ->  ( -.  A  <  B  ->  ( A  =  B  \/  B  <  A ) ) )
6111, 12, 60syl2anb 465 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  A  <  B  -> 
( A  =  B  \/  B  <  A
) ) )
6210, 61impbid 183 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  =  B  \/  B  <  A
)  <->  -.  A  <  B ) )
6362con2bid 319 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 176    \/ wo 357    /\ wa 358    \/ w3o 933    = wceq 1625    e. wcel 1686   class class class wbr 4025   RRcr 8738    +oocpnf 8866    -oocmnf 8867   RR*cxr 8868    < clt 8869
This theorem is referenced by:  xrltso  10477  xrleloe  10480  xrltlen  10482
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-pre-lttri 8813  ax-pre-lttrn 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874
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