Theorem xrlttri 11972

Description: Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 10010 or axlttri 10109. (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

Step	Hyp	Ref	Expression
1		xrltnr 11953	. . . . . . . 8 |- ( A e. RR* -> -. A < A )
2	1	adantr 481	. . . . . . 7 |- ( ( A e. RR* /\ A = B ) -> -. A < A )
3		breq2 4657	. . . . . . . 8 |- ( A = B -> ( A < A <-> A < B ) )
4	3	adantl 482	. . . . . . 7 |- ( ( A e. RR* /\ A = B ) -> ( A < A <-> A < B ) )
5	2, 4	mtbid 314	. . . . . 6 |- ( ( A e. RR* /\ A = B ) -> -. A < B )
6	5	ex 450	. . . . 5 |- ( A e. RR* -> ( A = B -> -. A < B ) )
7	6	adantr 481	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( A = B -> -. A < B ) )
8		xrltnsym 11970	. . . . 5 |- ( ( B e. RR* /\ A e. RR* ) -> ( B < A -> -. A < B ) )
9	8	ancoms 469	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( B < A -> -. A < B ) )
10	7, 9	jaod 395	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A = B \/ B < A ) -> -. A < B ) )
11		elxr 11950	. . . 4 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
12		elxr 11950	. . . 4 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
13		axlttri 10109	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -. ( A = B \/ B < A ) ) )
14	13	biimprd 238	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( -. ( A = B \/ B < A ) -> A < B ) )
15	14	con1d 139	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( -. A < B -> ( A = B \/ B < A ) ) )
16		ltpnf 11954	. . . . . . . . 9 |- ( A e. RR -> A < +oo )
17	16	adantr 481	. . . . . . . 8 |- ( ( A e. RR /\ B = +oo ) -> A < +oo )
18		breq2 4657	. . . . . . . . 9 |- ( B = +oo -> ( A < B <-> A < +oo ) )
19	18	adantl 482	. . . . . . . 8 |- ( ( A e. RR /\ B = +oo ) -> ( A < B <-> A < +oo ) )
20	17, 19	mpbird 247	. . . . . . 7 |- ( ( A e. RR /\ B = +oo ) -> A < B )
21	20	pm2.24d 147	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( -. A < B -> ( A = B \/ B < A ) ) )
22		mnflt 11957	. . . . . . . . . 10 |- ( A e. RR -> -oo < A )
23	22	adantr 481	. . . . . . . . 9 |- ( ( A e. RR /\ B = -oo ) -> -oo < A )
24		breq1 4656	. . . . . . . . . 10 |- ( B = -oo -> ( B < A <-> -oo < A ) )
25	24	adantl 482	. . . . . . . . 9 |- ( ( A e. RR /\ B = -oo ) -> ( B < A <-> -oo < A ) )
26	23, 25	mpbird 247	. . . . . . . 8 |- ( ( A e. RR /\ B = -oo ) -> B < A )
27	26	olcd 408	. . . . . . 7 |- ( ( A e. RR /\ B = -oo ) -> ( A = B \/ B < A ) )
28	27	a1d 25	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( -. A < B -> ( A = B \/ B < A ) ) )
29	15, 21, 28	3jaodan 1394	. . . . 5 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( -. A < B -> ( A = B \/ B < A ) ) )
30		ltpnf 11954	. . . . . . . . . 10 |- ( B e. RR -> B < +oo )
31	30	adantl 482	. . . . . . . . 9 |- ( ( A = +oo /\ B e. RR ) -> B < +oo )
32		breq2 4657	. . . . . . . . . 10 |- ( A = +oo -> ( B < A <-> B < +oo ) )
33	32	adantr 481	. . . . . . . . 9 |- ( ( A = +oo /\ B e. RR ) -> ( B < A <-> B < +oo ) )
34	31, 33	mpbird 247	. . . . . . . 8 |- ( ( A = +oo /\ B e. RR ) -> B < A )
35	34	olcd 408	. . . . . . 7 |- ( ( A = +oo /\ B e. RR ) -> ( A = B \/ B < A ) )
36	35	a1d 25	. . . . . 6 |- ( ( A = +oo /\ B e. RR ) -> ( -. A < B -> ( A = B \/ B < A ) ) )
37		eqtr3 2643	. . . . . . . 8 |- ( ( A = +oo /\ B = +oo ) -> A = B )
38	37	orcd 407	. . . . . . 7 |- ( ( A = +oo /\ B = +oo ) -> ( A = B \/ B < A ) )
39	38	a1d 25	. . . . . 6 |- ( ( A = +oo /\ B = +oo ) -> ( -. A < B -> ( A = B \/ B < A ) ) )
40		mnfltpnf 11960	. . . . . . . . . 10 |- -oo < +oo
41		breq12 4658	. . . . . . . . . 10 |- ( ( B = -oo /\ A = +oo ) -> ( B < A <-> -oo < +oo ) )
42	40, 41	mpbiri 248	. . . . . . . . 9 |- ( ( B = -oo /\ A = +oo ) -> B < A )
43	42	ancoms 469	. . . . . . . 8 |- ( ( A = +oo /\ B = -oo ) -> B < A )
44	43	olcd 408	. . . . . . 7 |- ( ( A = +oo /\ B = -oo ) -> ( A = B \/ B < A ) )
45	44	a1d 25	. . . . . 6 |- ( ( A = +oo /\ B = -oo ) -> ( -. A < B -> ( A = B \/ B < A ) ) )
46	36, 39, 45	3jaodan 1394	. . . . 5 |- ( ( A = +oo /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( -. A < B -> ( A = B \/ B < A ) ) )
47		mnflt 11957	. . . . . . . . 9 |- ( B e. RR -> -oo < B )
48	47	adantl 482	. . . . . . . 8 |- ( ( A = -oo /\ B e. RR ) -> -oo < B )
49		breq1 4656	. . . . . . . . 9 |- ( A = -oo -> ( A < B <-> -oo < B ) )
50	49	adantr 481	. . . . . . . 8 |- ( ( A = -oo /\ B e. RR ) -> ( A < B <-> -oo < B ) )
51	48, 50	mpbird 247	. . . . . . 7 |- ( ( A = -oo /\ B e. RR ) -> A < B )
52	51	pm2.24d 147	. . . . . 6 |- ( ( A = -oo /\ B e. RR ) -> ( -. A < B -> ( A = B \/ B < A ) ) )
53		breq12 4658	. . . . . . . 8 |- ( ( A = -oo /\ B = +oo ) -> ( A < B <-> -oo < +oo ) )
54	40, 53	mpbiri 248	. . . . . . 7 |- ( ( A = -oo /\ B = +oo ) -> A < B )
55	54	pm2.24d 147	. . . . . 6 |- ( ( A = -oo /\ B = +oo ) -> ( -. A < B -> ( A = B \/ B < A ) ) )
56		eqtr3 2643	. . . . . . . 8 |- ( ( A = -oo /\ B = -oo ) -> A = B )
57	56	orcd 407	. . . . . . 7 |- ( ( A = -oo /\ B = -oo ) -> ( A = B \/ B < A ) )
58	57	a1d 25	. . . . . 6 |- ( ( A = -oo /\ B = -oo ) -> ( -. A < B -> ( A = B \/ B < A ) ) )
59	52, 55, 58	3jaodan 1394	. . . . 5 |- ( ( A = -oo /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( -. A < B -> ( A = B \/ B < A ) ) )
60	29, 46, 59	3jaoian 1393	. . . 4 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( -. A < B -> ( A = B \/ B < A ) ) )
61	11, 12, 60	syl2anb 496	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( -. A < B -> ( A = B \/ B < A ) ) )
62	10, 61	impbid 202	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A = B \/ B < A ) <-> -. A < B ) )
63	62	con2bid 344	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> -. ( A = B \/ B < A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   class class class wbr 4653   RRcr 9935   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079

This theorem is referenced by:  xrltso  11974  xrleloe  11977  xrltlen  11979