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Theorem xrlttrit 5525
Description: Ordering on the extended reals satisfies strict trichotomy.
Assertion
Ref Expression
xrlttrit |- ((A e. RR* /\ B e. RR*) -> (A < B <-> -. (A = B \/ B < A)))
Proof of Theorem xrlttrit
StepHypRef Expression
1 xrltnrt 5514 . . . . . . . 8 |- (A e. RR* -> -. A < A)
21adantr 389 . . . . . . 7 |- ((A e. RR* /\ A = B) -> -. A < A)
3 breq2 2613 . . . . . . . 8 |- (A = B -> (A < A <-> A < B))
43adantl 388 . . . . . . 7 |- ((A e. RR* /\ A = B) -> (A < A <-> A < B))
52, 4mtbid 712 . . . . . 6 |- ((A e. RR* /\ A = B) -> -. A < B)
65ex 373 . . . . 5 |- (A e. RR* -> (A = B -> -. A < B))
76adantr 389 . . . 4 |- ((A e. RR* /\ B e. RR*) -> (A = B -> -. A < B))
8 xrltnsymt 5523 . . . . 5 |- ((B e. RR* /\ A e. RR*) -> (B < A -> -. A < B))
98ancoms 436 . . . 4 |- ((A e. RR* /\ B e. RR*) -> (B < A -> -. A < B))
107, 9jaod 424 . . 3 |- ((A e. RR* /\ B e. RR*) -> ((A = B \/ B < A) -> -. A < B))
11 axlttri 5475 . . . . . . . 8 |- ((A e. RR /\ B e. RR) -> (A < B <-> -. (A = B \/ B < A)))
1211biimprd 154 . . . . . . 7 |- ((A e. RR /\ B e. RR) -> (-. (A = B \/ B < A) -> A < B))
1312con1d 93 . . . . . 6 |- ((A e. RR /\ B e. RR) -> (-. A < B -> (A = B \/ B < A)))
14 ltpnft 5515 . . . . . . . . 9 |- (A e. RR -> A < +oo)
1514adantr 389 . . . . . . . 8 |- ((A e. RR /\ B = +oo) -> A < +oo)
16 breq2 2613 . . . . . . . . 9 |- (B = +oo -> (A < B <-> A < +oo))
1716adantl 388 . . . . . . . 8 |- ((A e. RR /\ B = +oo) -> (A < B <-> A < +oo))
1815, 17mpbird 196 . . . . . . 7 |- ((A e. RR /\ B = +oo) -> A < B)
1918pm2.24d 105 . . . . . 6 |- ((A e. RR /\ B = +oo) -> (-. A < B -> (A = B \/ B < A)))
20 mnfltt 5516 . . . . . . . . . 10 |- (A e. RR -> -oo < A)
2120adantr 389 . . . . . . . . 9 |- ((A e. RR /\ B = -oo) -> -oo < A)
22 breq1 2612 . . . . . . . . . 10 |- (B = -oo -> (B < A <-> -oo < A))
2322adantl 388 . . . . . . . . 9 |- ((A e. RR /\ B = -oo) -> (B < A <-> -oo < A))
2421, 23mpbird 196 . . . . . . . 8 |- ((A e. RR /\ B = -oo) -> B < A)
2524olcd 273 . . . . . . 7 |- ((A e. RR /\ B = -oo) -> (A = B \/ B < A))
2625a1d 12 . . . . . 6 |- ((A e. RR /\ B = -oo) -> (-. A < B -> (A = B \/ B < A)))
2713, 19, 263jaodan 887 . . . . 5 |- ((A e. RR /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (-. A < B -> (A = B \/ B < A)))
28 ltpnft 5515 . . . . . . . . . 10 |- (B e. RR -> B < +oo)
2928adantl 388 . . . . . . . . 9 |- ((A = +oo /\ B e. RR) -> B < +oo)
30 breq2 2613 . . . . . . . . . 10 |- (A = +oo -> (B < A <-> B < +oo))
3130adantr 389 . . . . . . . . 9 |- ((A = +oo /\ B e. RR) -> (B < A <-> B < +oo))
3229, 31mpbird 196 . . . . . . . 8 |- ((A = +oo /\ B e. RR) -> B < A)
3332olcd 273 . . . . . . 7 |- ((A = +oo /\ B e. RR) -> (A = B \/ B < A))
3433a1d 12 . . . . . 6 |- ((A = +oo /\ B e. RR) -> (-. A < B -> (A = B \/ B < A)))
35 eqtr3t 1486 . . . . . . . 8 |- ((A = +oo /\ B = +oo) -> A = B)
3635orcd 272 . . . . . . 7 |- ((A = +oo /\ B = +oo) -> (A = B \/ B < A))
3736a1d 12 . . . . . 6 |- ((A = +oo /\ B = +oo) -> (-. A < B -> (A = B \/ B < A)))
38 mnfltpnf 5517 . . . . . . . . . 10 |- -oo < +oo
39 breq12 2614 . . . . . . . . . 10 |- ((B = -oo /\ A = +oo) -> (B < A <-> -oo < +oo))
4038, 39mpbiri 194 . . . . . . . . 9 |- ((B = -oo /\ A = +oo) -> B < A)
4140ancoms 436 . . . . . . . 8 |- ((A = +oo /\ B = -oo) -> B < A)
4241olcd 273 . . . . . . 7 |- ((A = +oo /\ B = -oo) -> (A = B \/ B < A))
4342a1d 12 . . . . . 6 |- ((A = +oo /\ B = -oo) -> (-. A < B -> (A = B \/ B < A)))
4434, 37, 433jaodan 887 . . . . 5 |- ((A = +oo /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (-. A < B -> (A = B \/ B < A)))
45 mnfltt 5516 . . . . . . . . 9 |- (B e. RR -> -oo < B)
4645adantl 388 . . . . . . . 8 |- ((A = -oo /\ B e. RR) -> -oo < B)
47 breq1 2612 . . . . . . . . 9 |- (A = -oo -> (A < B <-> -oo < B))
4847adantr 389 . . . . . . . 8 |- ((A = -oo /\ B e. RR) -> (A < B <-> -oo < B))
4946, 48mpbird 196 . . . . . . 7 |- ((A = -oo /\ B e. RR) -> A < B)
5049pm2.24d 105 . . . . . 6 |- ((A = -oo /\ B e. RR) -> (-. A < B -> (A = B \/ B < A)))
51 breq12 2614 . . . . . . . 8 |- ((A = -oo /\ B = +oo) -> (A < B <-> -oo < +oo))
5238, 51mpbiri 194 . . . . . . 7 |- ((A = -oo /\ B = +oo) -> A < B)
5352pm2.24d 105 . . . . . 6 |- ((A = -oo /\ B = +oo) -> (-. A < B -> (A = B \/ B < A)))
54 eqtr3t 1486 . . . . . . . 8 |- ((A = -oo /\ B = -oo) -> A = B)
5554orcd 272 . . . . . . 7 |- ((A = -oo /\ B = -oo) -> (A = B \/ B < A))
5655a1d 12 . . . . . 6 |- ((A = -oo /\ B = -oo) -> (-. A < B -> (A = B \/ B < A)))
5750, 53, 563jaodan 887 . . . . 5 |- ((A = -oo /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (-. A < B -> (A = B \/ B < A)))
5827, 44, 573jaoian 886 . . . 4 |- (((A e. RR \/ A = +oo \/ A = -oo) /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (-. A < B -> (A = B \/ B < A)))
59 elxr 5508 . . . 4 |- (A e. RR* <-> (A e. RR \/ A = +oo \/ A = -oo))
60 elxr 5508 . . . 4 |- (B e. RR* <-> (B e. RR \/ B = +oo \/ B = -oo))
6158, 59, 60syl2anb 455 . . 3 |- ((A e. RR* /\ B e. RR*) -> (-. A < B -> (A = B \/ B < A)))
6210, 61impbid 514 . 2 |- ((A e. RR* /\ B e. RR*) -> ((A = B \/ B < A) <-> -. A < B))
6362con2bid 524 1 |- ((A e. RR* /\ B e. RR*) -> (A < B <-> -. (A = B \/ B < A)))
Colors of variables: wff set class
Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   \/ w3o 772   = wceq 953   e. wcel 955   class class class wbr 2609  RRcr 5205   +oocpnf 5455   -oocmnf 5456  RR*cxr 5457   < clt 5458
This theorem is referenced by:  xrltso 5527  xrleloet 5530
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-ltp 5062  df-enr 5138  df-nr 5139  df-ltr 5142  df-0r 5143  df-c 5212  df-r 5216  df-lt 5219  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462
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