Theorem xrlttrit 5533

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

Step	Hyp	Ref	Expression
1		xrltnrt 5522	. . . . . . . 8 |- (A e. RR* -> -. A < A)
2	1	adantr 389	. . . . . . 7 |- ((A e. RR* /\ A = B) -> -. A < A)
3		breq2 2618	. . . . . . . 8 |- (A = B -> (A < A <-> A < B))
4	3	adantl 388	. . . . . . 7 |- ((A e. RR* /\ A = B) -> (A < A <-> A < B))
5	2, 4	mtbid 713	. . . . . 6 |- ((A e. RR* /\ A = B) -> -. A < B)
6	5	ex 373	. . . . 5 |- (A e. RR* -> (A = B -> -. A < B))
7	6	adantr 389	. . . 4 |- ((A e. RR* /\ B e. RR*) -> (A = B -> -. A < B))
8		xrltnsymt 5531	. . . . 5 |- ((B e. RR* /\ A e. RR*) -> (B < A -> -. A < B))
9	8	ancoms 436	. . . 4 |- ((A e. RR* /\ B e. RR*) -> (B < A -> -. A < B))
10	7, 9	jaod 424	. . 3 |- ((A e. RR* /\ B e. RR*) -> ((A = B \/ B < A) -> -. A < B))
11		axlttri 5483	. . . . . . . 8 |- ((A e. RR /\ B e. RR) -> (A < B <-> -. (A = B \/ B < A)))
12	11	biimprd 154	. . . . . . 7 |- ((A e. RR /\ B e. RR) -> (-. (A = B \/ B < A) -> A < B))
13	12	con1d 93	. . . . . 6 |- ((A e. RR /\ B e. RR) -> (-. A < B -> (A = B \/ B < A)))
14		ltpnft 5523	. . . . . . . . 9 |- (A e. RR -> A < +oo)
15	14	adantr 389	. . . . . . . 8 |- ((A e. RR /\ B = +oo) -> A < +oo)
16		breq2 2618	. . . . . . . . 9 |- (B = +oo -> (A < B <-> A < +oo))
17	16	adantl 388	. . . . . . . 8 |- ((A e. RR /\ B = +oo) -> (A < B <-> A < +oo))
18	15, 17	mpbird 196	. . . . . . 7 |- ((A e. RR /\ B = +oo) -> A < B)
19	18	pm2.24d 105	. . . . . 6 |- ((A e. RR /\ B = +oo) -> (-. A < B -> (A = B \/ B < A)))
20		mnfltt 5524	. . . . . . . . . 10 |- (A e. RR -> -oo < A)
21	20	adantr 389	. . . . . . . . 9 |- ((A e. RR /\ B = -oo) -> -oo < A)
22		breq1 2617	. . . . . . . . . 10 |- (B = -oo -> (B < A <-> -oo < A))
23	22	adantl 388	. . . . . . . . 9 |- ((A e. RR /\ B = -oo) -> (B < A <-> -oo < A))
24	21, 23	mpbird 196	. . . . . . . 8 |- ((A e. RR /\ B = -oo) -> B < A)
25	24	olcd 273	. . . . . . 7 |- ((A e. RR /\ B = -oo) -> (A = B \/ B < A))
26	25	a1d 12	. . . . . 6 |- ((A e. RR /\ B = -oo) -> (-. A < B -> (A = B \/ B < A)))
27	13, 19, 26	3jaodan 888	. . . . 5 |- ((A e. RR /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (-. A < B -> (A = B \/ B < A)))
28		ltpnft 5523	. . . . . . . . . 10 |- (B e. RR -> B < +oo)
29	28	adantl 388	. . . . . . . . 9 |- ((A = +oo /\ B e. RR) -> B < +oo)
30		breq2 2618	. . . . . . . . . 10 |- (A = +oo -> (B < A <-> B < +oo))
31	30	adantr 389	. . . . . . . . 9 |- ((A = +oo /\ B e. RR) -> (B < A <-> B < +oo))
32	29, 31	mpbird 196	. . . . . . . 8 |- ((A = +oo /\ B e. RR) -> B < A)
33	32	olcd 273	. . . . . . 7 |- ((A = +oo /\ B e. RR) -> (A = B \/ B < A))
34	33	a1d 12	. . . . . 6 |- ((A = +oo /\ B e. RR) -> (-. A < B -> (A = B \/ B < A)))
35		eqtr3t 1491	. . . . . . . 8 |- ((A = +oo /\ B = +oo) -> A = B)
36	35	orcd 272	. . . . . . 7 |- ((A = +oo /\ B = +oo) -> (A = B \/ B < A))
37	36	a1d 12	. . . . . 6 |- ((A = +oo /\ B = +oo) -> (-. A < B -> (A = B \/ B < A)))
38		mnfltpnf 5525	. . . . . . . . . 10 |- -oo < +oo
39		breq12 2619	. . . . . . . . . 10 |- ((B = -oo /\ A = +oo) -> (B < A <-> -oo < +oo))
40	38, 39	mpbiri 194	. . . . . . . . 9 |- ((B = -oo /\ A = +oo) -> B < A)
41	40	ancoms 436	. . . . . . . 8 |- ((A = +oo /\ B = -oo) -> B < A)
42	41	olcd 273	. . . . . . 7 |- ((A = +oo /\ B = -oo) -> (A = B \/ B < A))
43	42	a1d 12	. . . . . 6 |- ((A = +oo /\ B = -oo) -> (-. A < B -> (A = B \/ B < A)))
44	34, 37, 43	3jaodan 888	. . . . 5 |- ((A = +oo /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (-. A < B -> (A = B \/ B < A)))
45		mnfltt 5524	. . . . . . . . 9 |- (B e. RR -> -oo < B)
46	45	adantl 388	. . . . . . . 8 |- ((A = -oo /\ B e. RR) -> -oo < B)
47		breq1 2617	. . . . . . . . 9 |- (A = -oo -> (A < B <-> -oo < B))
48	47	adantr 389	. . . . . . . 8 |- ((A = -oo /\ B e. RR) -> (A < B <-> -oo < B))
49	46, 48	mpbird 196	. . . . . . 7 |- ((A = -oo /\ B e. RR) -> A < B)
50	49	pm2.24d 105	. . . . . 6 |- ((A = -oo /\ B e. RR) -> (-. A < B -> (A = B \/ B < A)))
51		breq12 2619	. . . . . . . 8 |- ((A = -oo /\ B = +oo) -> (A < B <-> -oo < +oo))
52	38, 51	mpbiri 194	. . . . . . 7 |- ((A = -oo /\ B = +oo) -> A < B)
53	52	pm2.24d 105	. . . . . 6 |- ((A = -oo /\ B = +oo) -> (-. A < B -> (A = B \/ B < A)))
54		eqtr3t 1491	. . . . . . . 8 |- ((A = -oo /\ B = -oo) -> A = B)
55	54	orcd 272	. . . . . . 7 |- ((A = -oo /\ B = -oo) -> (A = B \/ B < A))
56	55	a1d 12	. . . . . 6 |- ((A = -oo /\ B = -oo) -> (-. A < B -> (A = B \/ B < A)))
57	50, 53, 56	3jaodan 888	. . . . 5 |- ((A = -oo /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (-. A < B -> (A = B \/ B < A)))
58	27, 44, 57	3jaoian 887	. . . 4 |- (((A e. RR \/ A = +oo \/ A = -oo) /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (-. A < B -> (A = B \/ B < A)))
59		elxr 5516	. . . 4 |- (A e. RR* <-> (A e. RR \/ A = +oo \/ A = -oo))
60		elxr 5516	. . . 4 |- (B e. RR* <-> (B e. RR \/ B = +oo \/ B = -oo))
61	58, 59, 60	syl2anb 455	. . 3 |- ((A e. RR* /\ B e. RR*) -> (-. A < B -> (A = B \/ B < A)))
62	10, 61	impbid 515	. 2 |- ((A e. RR* /\ B e. RR*) -> ((A = B \/ B < A) <-> -. A < B))
63	62	con2bid 525	1 |- ((A e. RR* /\ B e. RR*) -> (A < B <-> -. (A = B \/ B < A)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   \/ w3o 773   = wceq 954   e. wcel 956   class class class wbr 2614  RRcr 5213   +oocpnf 5463   -oocmnf 5464  RR*cxr 5465   < clt 5466

This theorem is referenced by:  xrltso 5535  xrleloet 5538

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-en 4357  df-dom 4358  df-sdom 4359  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-ltp 5070  df-enr 5146  df-nr 5147  df-ltr 5150  df-0r 5151  df-c 5220  df-r 5224  df-lt 5227  df-pnf 5467  df-mnf 5468  df-xr 5469  df-ltxr 5470

Copyright terms: Public domain