Theorem xrlttri 6727

Description: Ordering on the extended reals satisfies strict trichotomy.

Assertion

Ref	Expression
xrlttri	|- ((A e. RR* /\ B e. RR*) -> (A < B <-> -. (A = B \/ B < A)))

Proof of Theorem xrlttri

Step	Hyp	Ref	Expression
1		xrltnr 6716	. . . . . . . 8 |- (A e. RR* -> -. A < A)
2	1	adantr 425	. . . . . . 7 |- ((A e. RR* /\ A = B) -> -. A < A)
3		breq2 3342	. . . . . . . 8 |- (A = B -> (A < A <-> A < B))
4	3	adantl 424	. . . . . . 7 |- ((A e. RR* /\ A = B) -> (A < A <-> A < B))
5	2, 4	mtbid 782	. . . . . 6 |- ((A e. RR* /\ A = B) -> -. A < B)
6	5	ex 402	. . . . 5 |- (A e. RR* -> (A = B -> -. A < B))
7	6	adantr 425	. . . 4 |- ((A e. RR* /\ B e. RR*) -> (A = B -> -. A < B))
8		xrltnsym 6725	. . . . 5 |- ((B e. RR* /\ A e. RR*) -> (B < A -> -. A < B))
9	8	ancoms 484	. . . 4 |- ((A e. RR* /\ B e. RR*) -> (B < A -> -. A < B))
10	7, 9	jaod 469	. . 3 |- ((A e. RR* /\ B e. RR*) -> ((A = B \/ B < A) -> -. A < B))
11		axlttri 6672	. . . . . . . 8 |- ((A e. RR /\ B e. RR) -> (A < B <-> -. (A = B \/ B < A)))
12	11	biimprd 171	. . . . . . 7 |- ((A e. RR /\ B e. RR) -> (-. (A = B \/ B < A) -> A < B))
13	12	con1d 109	. . . . . 6 |- ((A e. RR /\ B e. RR) -> (-. A < B -> (A = B \/ B < A)))
14		ltpnf 6717	. . . . . . . . 9 |- (A e. RR -> A < +oo)
15	14	adantr 425	. . . . . . . 8 |- ((A e. RR /\ B = +oo) -> A < +oo)
16		breq2 3342	. . . . . . . . 9 |- (B = +oo -> (A < B <-> A < +oo))
17	16	adantl 424	. . . . . . . 8 |- ((A e. RR /\ B = +oo) -> (A < B <-> A < +oo))
18	15, 17	mpbird 213	. . . . . . 7 |- ((A e. RR /\ B = +oo) -> A < B)
19	18	pm2.24d 120	. . . . . 6 |- ((A e. RR /\ B = +oo) -> (-. A < B -> (A = B \/ B < A)))
20		mnflt 6718	. . . . . . . . . 10 |- (A e. RR -> -oo < A)
21	20	adantr 425	. . . . . . . . 9 |- ((A e. RR /\ B = -oo) -> -oo < A)
22		breq1 3341	. . . . . . . . . 10 |- (B = -oo -> (B < A <-> -oo < A))
23	22	adantl 424	. . . . . . . . 9 |- ((A e. RR /\ B = -oo) -> (B < A <-> -oo < A))
24	21, 23	mpbird 213	. . . . . . . 8 |- ((A e. RR /\ B = -oo) -> B < A)
25	24	olcd 295	. . . . . . 7 |- ((A e. RR /\ B = -oo) -> (A = B \/ B < A))
26	25	a1d 15	. . . . . 6 |- ((A e. RR /\ B = -oo) -> (-. A < B -> (A = B \/ B < A)))
27	13, 19, 26	3jaodan 1163	. . . . 5 |- ((A e. RR /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (-. A < B -> (A = B \/ B < A)))
28		ltpnf 6717	. . . . . . . . . 10 |- (B e. RR -> B < +oo)
29	28	adantl 424	. . . . . . . . 9 |- ((A = +oo /\ B e. RR) -> B < +oo)
30		breq2 3342	. . . . . . . . . 10 |- (A = +oo -> (B < A <-> B < +oo))
31	30	adantr 425	. . . . . . . . 9 |- ((A = +oo /\ B e. RR) -> (B < A <-> B < +oo))
32	29, 31	mpbird 213	. . . . . . . 8 |- ((A = +oo /\ B e. RR) -> B < A)
33	32	olcd 295	. . . . . . 7 |- ((A = +oo /\ B e. RR) -> (A = B \/ B < A))
34	33	a1d 15	. . . . . 6 |- ((A = +oo /\ B e. RR) -> (-. A < B -> (A = B \/ B < A)))
35		eqtr3 1907	. . . . . . . 8 |- ((A = +oo /\ B = +oo) -> A = B)
36	35	orcd 294	. . . . . . 7 |- ((A = +oo /\ B = +oo) -> (A = B \/ B < A))
37	36	a1d 15	. . . . . 6 |- ((A = +oo /\ B = +oo) -> (-. A < B -> (A = B \/ B < A)))
38		mnfltpnf 6719	. . . . . . . . . 10 |- -oo < +oo
39		breq12 3343	. . . . . . . . . 10 |- ((B = -oo /\ A = +oo) -> (B < A <-> -oo < +oo))
40	38, 39	mpbiri 211	. . . . . . . . 9 |- ((B = -oo /\ A = +oo) -> B < A)
41	40	ancoms 484	. . . . . . . 8 |- ((A = +oo /\ B = -oo) -> B < A)
42	41	olcd 295	. . . . . . 7 |- ((A = +oo /\ B = -oo) -> (A = B \/ B < A))
43	42	a1d 15	. . . . . 6 |- ((A = +oo /\ B = -oo) -> (-. A < B -> (A = B \/ B < A)))
44	34, 37, 43	3jaodan 1163	. . . . 5 |- ((A = +oo /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (-. A < B -> (A = B \/ B < A)))
45		mnflt 6718	. . . . . . . . 9 |- (B e. RR -> -oo < B)
46	45	adantl 424	. . . . . . . 8 |- ((A = -oo /\ B e. RR) -> -oo < B)
47		breq1 3341	. . . . . . . . 9 |- (A = -oo -> (A < B <-> -oo < B))
48	47	adantr 425	. . . . . . . 8 |- ((A = -oo /\ B e. RR) -> (A < B <-> -oo < B))
49	46, 48	mpbird 213	. . . . . . 7 |- ((A = -oo /\ B e. RR) -> A < B)
50	49	pm2.24d 120	. . . . . 6 |- ((A = -oo /\ B e. RR) -> (-. A < B -> (A = B \/ B < A)))
51		breq12 3343	. . . . . . . 8 |- ((A = -oo /\ B = +oo) -> (A < B <-> -oo < +oo))
52	38, 51	mpbiri 211	. . . . . . 7 |- ((A = -oo /\ B = +oo) -> A < B)
53	52	pm2.24d 120	. . . . . 6 |- ((A = -oo /\ B = +oo) -> (-. A < B -> (A = B \/ B < A)))
54		eqtr3 1907	. . . . . . . 8 |- ((A = -oo /\ B = -oo) -> A = B)
55	54	orcd 294	. . . . . . 7 |- ((A = -oo /\ B = -oo) -> (A = B \/ B < A))
56	55	a1d 15	. . . . . 6 |- ((A = -oo /\ B = -oo) -> (-. A < B -> (A = B \/ B < A)))
57	50, 53, 56	3jaodan 1163	. . . . 5 |- ((A = -oo /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (-. A < B -> (A = B \/ B < A)))
58	27, 44, 57	3jaoian 1162	. . . 4 |- (((A e. RR \/ A = +oo \/ A = -oo) /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (-. A < B -> (A = B \/ B < A)))
59		elxr 6706	. . . 4 |- (A e. RR* <-> (A e. RR \/ A = +oo \/ A = -oo))
60		elxr 6706	. . . 4 |- (B e. RR* <-> (B e. RR \/ B = +oo \/ B = -oo))
61	58, 59, 60	syl2anb 504	. . 3 |- ((A e. RR* /\ B e. RR*) -> (-. A < B -> (A = B \/ B < A)))
62	10, 61	impbid 574	. 2 |- ((A e. RR* /\ B e. RR*) -> ((A = B \/ B < A) <-> -. A < B))
63	62	con2bid 585	1 |- ((A e. RR* /\ B e. RR*) -> (A < B <-> -. (A = B \/ B < A)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   \/ w3o 857   = wceq 1298   e. wcel 1300   class class class wbr 3338  RRcr 6385   +oocpnf 6650   -oocmnf 6651  RR*cxr 6652   < clt 6653

This theorem is referenced by:  xrltso 6729  xrleloe 6732

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-ltp 6242  df-enr 6318  df-nr 6319  df-ltr 6322  df-0r 6323  df-c 6392  df-r 6396  df-lt 6399  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657

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