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Theorem xrlttri3 9000
Description: Extended real version of lttri3 7310. (Contributed by NM, 9-Feb-2006.)
Assertion
Ref Expression
xrlttri3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )

Proof of Theorem xrlttri3
StepHypRef Expression
1 elxr 8980 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 8980 . 2  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 lttri3 7310 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
43ancoms 264 . . . . 5  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
5 renepnf 7280 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  =/= +oo )
65adantr 270 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  B  =/= +oo )
7 neeq2 2263 . . . . . . . . . 10  |-  ( A  = +oo  ->  ( B  =/=  A  <->  B  =/= +oo ) )
87adantl 271 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  ( B  =/=  A  <->  B  =/= +oo ) )
96, 8mpbird 165 . . . . . . . 8  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  B  =/=  A )
109necomd 2335 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  A  =/=  B )
1110neneqd 2270 . . . . . 6  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  -.  A  =  B )
12 ltpnf 8984 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  < +oo )
1312adantr 270 . . . . . . . 8  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  B  < +oo )
14 breq2 3809 . . . . . . . . 9  |-  ( A  = +oo  ->  ( B  <  A  <->  B  < +oo ) )
1514adantl 271 . . . . . . . 8  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  ( B  <  A  <->  B  < +oo ) )
1613, 15mpbird 165 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  B  <  A )
17 notnot 592 . . . . . . . . 9  |-  ( ( A  <  B  \/  B  <  A )  ->  -.  -.  ( A  < 
B  \/  B  < 
A ) )
1817olcs 688 . . . . . . . 8  |-  ( B  <  A  ->  -.  -.  ( A  <  B  \/  B  <  A ) )
19 ioran 702 . . . . . . . 8  |-  ( -.  ( A  <  B  \/  B  <  A )  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) )
2018, 19sylnib 634 . . . . . . 7  |-  ( B  <  A  ->  -.  ( -.  A  <  B  /\  -.  B  < 
A ) )
2116, 20syl 14 . . . . . 6  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  -.  ( -.  A  <  B  /\  -.  B  <  A ) )
2211, 212falsed 651 . . . . 5  |-  ( ( B  e.  RR  /\  A  = +oo )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
23 renemnf 7281 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  =/= -oo )
2423adantr 270 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  = -oo )  ->  B  =/= -oo )
25 neeq2 2263 . . . . . . . . . 10  |-  ( A  = -oo  ->  ( B  =/=  A  <->  B  =/= -oo ) )
2625adantl 271 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  = -oo )  ->  ( B  =/=  A  <->  B  =/= -oo ) )
2724, 26mpbird 165 . . . . . . . 8  |-  ( ( B  e.  RR  /\  A  = -oo )  ->  B  =/=  A )
2827necomd 2335 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  = -oo )  ->  A  =/=  B )
2928neneqd 2270 . . . . . 6  |-  ( ( B  e.  RR  /\  A  = -oo )  ->  -.  A  =  B )
30 mnflt 8986 . . . . . . . . 9  |-  ( B  e.  RR  -> -oo  <  B )
3130adantr 270 . . . . . . . 8  |-  ( ( B  e.  RR  /\  A  = -oo )  -> -oo  <  B )
32 breq1 3808 . . . . . . . . 9  |-  ( A  = -oo  ->  ( A  <  B  <-> -oo  <  B
) )
3332adantl 271 . . . . . . . 8  |-  ( ( B  e.  RR  /\  A  = -oo )  ->  ( A  <  B  <-> -oo 
<  B ) )
3431, 33mpbird 165 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  = -oo )  ->  A  <  B )
35 orc 666 . . . . . . 7  |-  ( A  <  B  ->  ( A  <  B  \/  B  <  A ) )
36 oranim 841 . . . . . . 7  |-  ( ( A  <  B  \/  B  <  A )  ->  -.  ( -.  A  < 
B  /\  -.  B  <  A ) )
3734, 35, 363syl 17 . . . . . 6  |-  ( ( B  e.  RR  /\  A  = -oo )  ->  -.  ( -.  A  <  B  /\  -.  B  <  A ) )
3829, 372falsed 651 . . . . 5  |-  ( ( B  e.  RR  /\  A  = -oo )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
394, 22, 383jaodan 1238 . . . 4  |-  ( ( B  e.  RR  /\  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )
4039ancoms 264 . . 3  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  e.  RR )  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )
41 renepnf 7280 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  =/= +oo )
4241adantl 271 . . . . . . . 8  |-  ( ( B  = +oo  /\  A  e.  RR )  ->  A  =/= +oo )
43 neeq2 2263 . . . . . . . . 9  |-  ( B  = +oo  ->  ( A  =/=  B  <->  A  =/= +oo ) )
4443adantr 270 . . . . . . . 8  |-  ( ( B  = +oo  /\  A  e.  RR )  ->  ( A  =/=  B  <->  A  =/= +oo ) )
4542, 44mpbird 165 . . . . . . 7  |-  ( ( B  = +oo  /\  A  e.  RR )  ->  A  =/=  B )
4645neneqd 2270 . . . . . 6  |-  ( ( B  = +oo  /\  A  e.  RR )  ->  -.  A  =  B )
47 ltpnf 8984 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  < +oo )
4847adantl 271 . . . . . . . 8  |-  ( ( B  = +oo  /\  A  e.  RR )  ->  A  < +oo )
49 breq2 3809 . . . . . . . . 9  |-  ( B  = +oo  ->  ( A  <  B  <->  A  < +oo ) )
5049adantr 270 . . . . . . . 8  |-  ( ( B  = +oo  /\  A  e.  RR )  ->  ( A  <  B  <->  A  < +oo ) )
5148, 50mpbird 165 . . . . . . 7  |-  ( ( B  = +oo  /\  A  e.  RR )  ->  A  <  B )
5251, 35, 363syl 17 . . . . . 6  |-  ( ( B  = +oo  /\  A  e.  RR )  ->  -.  ( -.  A  <  B  /\  -.  B  <  A ) )
5346, 522falsed 651 . . . . 5  |-  ( ( B  = +oo  /\  A  e.  RR )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
54 eqtr3 2102 . . . . . . 7  |-  ( ( B  = +oo  /\  A  = +oo )  ->  B  =  A )
5554eqcomd 2088 . . . . . 6  |-  ( ( B  = +oo  /\  A  = +oo )  ->  A  =  B )
56 pnfxr 7285 . . . . . . . . 9  |- +oo  e.  RR*
57 xrltnr 8983 . . . . . . . . 9  |-  ( +oo  e.  RR*  ->  -. +oo  < +oo )
5856, 57ax-mp 7 . . . . . . . 8  |-  -. +oo  < +oo
59 breq12 3810 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  = +oo )  ->  ( A  <  B  <-> +oo 
< +oo ) )
6059ancoms 264 . . . . . . . 8  |-  ( ( B  = +oo  /\  A  = +oo )  ->  ( A  <  B  <-> +oo 
< +oo ) )
6158, 60mtbiri 633 . . . . . . 7  |-  ( ( B  = +oo  /\  A  = +oo )  ->  -.  A  <  B
)
62 breq12 3810 . . . . . . . 8  |-  ( ( B  = +oo  /\  A  = +oo )  ->  ( B  <  A  <-> +oo 
< +oo ) )
6358, 62mtbiri 633 . . . . . . 7  |-  ( ( B  = +oo  /\  A  = +oo )  ->  -.  B  <  A
)
6461, 63jca 300 . . . . . 6  |-  ( ( B  = +oo  /\  A  = +oo )  ->  ( -.  A  < 
B  /\  -.  B  <  A ) )
6555, 642thd 173 . . . . 5  |-  ( ( B  = +oo  /\  A  = +oo )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
66 mnfnepnf 7288 . . . . . . . . 9  |- -oo  =/= +oo
67 eqeq12 2095 . . . . . . . . . 10  |-  ( ( A  = -oo  /\  B  = +oo )  ->  ( A  =  B  <-> -oo  = +oo )
)
6867necon3bid 2290 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  = +oo )  ->  ( A  =/=  B  <-> -oo 
=/= +oo ) )
6966, 68mpbiri 166 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  = +oo )  ->  A  =/=  B )
7069ancoms 264 . . . . . . 7  |-  ( ( B  = +oo  /\  A  = -oo )  ->  A  =/=  B )
7170neneqd 2270 . . . . . 6  |-  ( ( B  = +oo  /\  A  = -oo )  ->  -.  A  =  B )
72 mnfltpnf 8988 . . . . . . . . 9  |- -oo  < +oo
73 breq12 3810 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  = +oo )  ->  ( A  <  B  <-> -oo 
< +oo ) )
7472, 73mpbiri 166 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  = +oo )  ->  A  <  B )
7574ancoms 264 . . . . . . 7  |-  ( ( B  = +oo  /\  A  = -oo )  ->  A  <  B )
7675, 35, 363syl 17 . . . . . 6  |-  ( ( B  = +oo  /\  A  = -oo )  ->  -.  ( -.  A  <  B  /\  -.  B  <  A ) )
7771, 762falsed 651 . . . . 5  |-  ( ( B  = +oo  /\  A  = -oo )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
7853, 65, 773jaodan 1238 . . . 4  |-  ( ( B  = +oo  /\  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )
7978ancoms 264 . . 3  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  = +oo )  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )
80 renemnf 7281 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  =/= -oo )
8180adantl 271 . . . . . . . 8  |-  ( ( B  = -oo  /\  A  e.  RR )  ->  A  =/= -oo )
82 neeq2 2263 . . . . . . . . 9  |-  ( B  = -oo  ->  ( A  =/=  B  <->  A  =/= -oo ) )
8382adantr 270 . . . . . . . 8  |-  ( ( B  = -oo  /\  A  e.  RR )  ->  ( A  =/=  B  <->  A  =/= -oo ) )
8481, 83mpbird 165 . . . . . . 7  |-  ( ( B  = -oo  /\  A  e.  RR )  ->  A  =/=  B )
8584neneqd 2270 . . . . . 6  |-  ( ( B  = -oo  /\  A  e.  RR )  ->  -.  A  =  B )
86 mnflt 8986 . . . . . . . . 9  |-  ( A  e.  RR  -> -oo  <  A )
8786adantl 271 . . . . . . . 8  |-  ( ( B  = -oo  /\  A  e.  RR )  -> -oo  <  A )
88 breq1 3808 . . . . . . . . 9  |-  ( B  = -oo  ->  ( B  <  A  <-> -oo  <  A
) )
8988adantr 270 . . . . . . . 8  |-  ( ( B  = -oo  /\  A  e.  RR )  ->  ( B  <  A  <-> -oo 
<  A ) )
9087, 89mpbird 165 . . . . . . 7  |-  ( ( B  = -oo  /\  A  e.  RR )  ->  B  <  A )
9190, 20syl 14 . . . . . 6  |-  ( ( B  = -oo  /\  A  e.  RR )  ->  -.  ( -.  A  <  B  /\  -.  B  <  A ) )
9285, 912falsed 651 . . . . 5  |-  ( ( B  = -oo  /\  A  e.  RR )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
9366neii 2251 . . . . . . . . . 10  |-  -. -oo  = +oo
94 eqeq12 2095 . . . . . . . . . 10  |-  ( ( B  = -oo  /\  A  = +oo )  ->  ( B  =  A  <-> -oo  = +oo )
)
9593, 94mtbiri 633 . . . . . . . . 9  |-  ( ( B  = -oo  /\  A  = +oo )  ->  -.  B  =  A )
9695neneqad 2328 . . . . . . . 8  |-  ( ( B  = -oo  /\  A  = +oo )  ->  B  =/=  A )
9796necomd 2335 . . . . . . 7  |-  ( ( B  = -oo  /\  A  = +oo )  ->  A  =/=  B )
9897neneqd 2270 . . . . . 6  |-  ( ( B  = -oo  /\  A  = +oo )  ->  -.  A  =  B )
99 breq12 3810 . . . . . . . 8  |-  ( ( B  = -oo  /\  A  = +oo )  ->  ( B  <  A  <-> -oo 
< +oo ) )
10072, 99mpbiri 166 . . . . . . 7  |-  ( ( B  = -oo  /\  A  = +oo )  ->  B  <  A )
101100, 20syl 14 . . . . . 6  |-  ( ( B  = -oo  /\  A  = +oo )  ->  -.  ( -.  A  <  B  /\  -.  B  <  A ) )
10298, 1012falsed 651 . . . . 5  |-  ( ( B  = -oo  /\  A  = +oo )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
103 eqtr3 2102 . . . . . . 7  |-  ( ( A  = -oo  /\  B  = -oo )  ->  A  =  B )
104103ancoms 264 . . . . . 6  |-  ( ( B  = -oo  /\  A  = -oo )  ->  A  =  B )
105 mnfxr 7289 . . . . . . . . 9  |- -oo  e.  RR*
106 xrltnr 8983 . . . . . . . . 9  |-  ( -oo  e.  RR*  ->  -. -oo  < -oo )
107105, 106ax-mp 7 . . . . . . . 8  |-  -. -oo  < -oo
108 breq12 3810 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( A  <  B  <-> -oo 
< -oo ) )
109108ancoms 264 . . . . . . . 8  |-  ( ( B  = -oo  /\  A  = -oo )  ->  ( A  <  B  <-> -oo 
< -oo ) )
110107, 109mtbiri 633 . . . . . . 7  |-  ( ( B  = -oo  /\  A  = -oo )  ->  -.  A  <  B
)
111 breq12 3810 . . . . . . . 8  |-  ( ( B  = -oo  /\  A  = -oo )  ->  ( B  <  A  <-> -oo 
< -oo ) )
112107, 111mtbiri 633 . . . . . . 7  |-  ( ( B  = -oo  /\  A  = -oo )  ->  -.  B  <  A
)
113110, 112jca 300 . . . . . 6  |-  ( ( B  = -oo  /\  A  = -oo )  ->  ( -.  A  < 
B  /\  -.  B  <  A ) )
114104, 1132thd 173 . . . . 5  |-  ( ( B  = -oo  /\  A  = -oo )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
11592, 102, 1143jaodan 1238 . . . 4  |-  ( ( B  = -oo  /\  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )
116115ancoms 264 . . 3  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  = -oo )  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )
11740, 79, 1163jaodan 1238 . 2  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  -> 
( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
1181, 2, 117syl2anb 285 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    /\ wa 102    <-> wb 103    \/ wo 662    \/ w3o 919    = wceq 1285    e. wcel 1434    =/= wne 2249   class class class wbr 3805   RRcr 7094   +oocpnf 7264   -oocmnf 7265   RR*cxr 7266    < clt 7267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-pre-ltirr 7202  ax-pre-apti 7205
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-xp 4397  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272
This theorem is referenced by:  xrletri3  9003  iccid  9076
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