Theorem xrlttri3 9704

Description: Extended real version of lttri3 7957. (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

Step	Hyp	Ref	Expression
1		elxr 9683	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 9683	. 2 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		lttri3 7957	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
4	3	ancoms 266	. . . . 5 |- ( ( B e. RR /\ A e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
5		renepnf 7925	. . . . . . . . . 10 |- ( B e. RR -> B =/= +oo )
6	5	adantr 274	. . . . . . . . 9 |- ( ( B e. RR /\ A = +oo ) -> B =/= +oo )
7		neeq2 2341	. . . . . . . . . 10 |- ( A = +oo -> ( B =/= A <-> B =/= +oo ) )
8	7	adantl 275	. . . . . . . . 9 |- ( ( B e. RR /\ A = +oo ) -> ( B =/= A <-> B =/= +oo ) )
9	6, 8	mpbird 166	. . . . . . . 8 |- ( ( B e. RR /\ A = +oo ) -> B =/= A )
10	9	necomd 2413	. . . . . . 7 |- ( ( B e. RR /\ A = +oo ) -> A =/= B )
11	10	neneqd 2348	. . . . . 6 |- ( ( B e. RR /\ A = +oo ) -> -. A = B )
12		ltpnf 9687	. . . . . . . . 9 |- ( B e. RR -> B < +oo )
13	12	adantr 274	. . . . . . . 8 |- ( ( B e. RR /\ A = +oo ) -> B < +oo )
14		breq2 3969	. . . . . . . . 9 |- ( A = +oo -> ( B < A <-> B < +oo ) )
15	14	adantl 275	. . . . . . . 8 |- ( ( B e. RR /\ A = +oo ) -> ( B < A <-> B < +oo ) )
16	13, 15	mpbird 166	. . . . . . 7 |- ( ( B e. RR /\ A = +oo ) -> B < A )
17		notnot 619	. . . . . . . . 9 |- ( ( A < B \/ B < A ) -> -. -. ( A < B \/ B < A ) )
18	17	olcs 726	. . . . . . . 8 |- ( B < A -> -. -. ( A < B \/ B < A ) )
19		ioran 742	. . . . . . . 8 |- ( -. ( A < B \/ B < A ) <-> ( -. A < B /\ -. B < A ) )
20	18, 19	sylnib 666	. . . . . . 7 |- ( B < A -> -. ( -. A < B /\ -. B < A ) )
21	16, 20	syl 14	. . . . . 6 |- ( ( B e. RR /\ A = +oo ) -> -. ( -. A < B /\ -. B < A ) )
22	11, 21	2falsed 692	. . . . 5 |- ( ( B e. RR /\ A = +oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
23		renemnf 7926	. . . . . . . . . 10 |- ( B e. RR -> B =/= -oo )
24	23	adantr 274	. . . . . . . . 9 |- ( ( B e. RR /\ A = -oo ) -> B =/= -oo )
25		neeq2 2341	. . . . . . . . . 10 |- ( A = -oo -> ( B =/= A <-> B =/= -oo ) )
26	25	adantl 275	. . . . . . . . 9 |- ( ( B e. RR /\ A = -oo ) -> ( B =/= A <-> B =/= -oo ) )
27	24, 26	mpbird 166	. . . . . . . 8 |- ( ( B e. RR /\ A = -oo ) -> B =/= A )
28	27	necomd 2413	. . . . . . 7 |- ( ( B e. RR /\ A = -oo ) -> A =/= B )
29	28	neneqd 2348	. . . . . 6 |- ( ( B e. RR /\ A = -oo ) -> -. A = B )
30		mnflt 9690	. . . . . . . . 9 |- ( B e. RR -> -oo < B )
31	30	adantr 274	. . . . . . . 8 |- ( ( B e. RR /\ A = -oo ) -> -oo < B )
32		breq1 3968	. . . . . . . . 9 |- ( A = -oo -> ( A < B <-> -oo < B ) )
33	32	adantl 275	. . . . . . . 8 |- ( ( B e. RR /\ A = -oo ) -> ( A < B <-> -oo < B ) )
34	31, 33	mpbird 166	. . . . . . 7 |- ( ( B e. RR /\ A = -oo ) -> A < B )
35		orc 702	. . . . . . 7 |- ( A < B -> ( A < B \/ B < A ) )
36		oranim 771	. . . . . . 7 |- ( ( A < B \/ B < A ) -> -. ( -. A < B /\ -. B < A ) )
37	34, 35, 36	3syl 17	. . . . . 6 |- ( ( B e. RR /\ A = -oo ) -> -. ( -. A < B /\ -. B < A ) )
38	29, 37	2falsed 692	. . . . 5 |- ( ( B e. RR /\ A = -oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
39	4, 22, 38	3jaodan 1288	. . . 4 |- ( ( B e. RR /\ ( A e. RR \/ A = +oo \/ A = -oo ) ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
40	39	ancoms 266	. . 3 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
41		renepnf 7925	. . . . . . . . 9 |- ( A e. RR -> A =/= +oo )
42	41	adantl 275	. . . . . . . 8 |- ( ( B = +oo /\ A e. RR ) -> A =/= +oo )
43		neeq2 2341	. . . . . . . . 9 |- ( B = +oo -> ( A =/= B <-> A =/= +oo ) )
44	43	adantr 274	. . . . . . . 8 |- ( ( B = +oo /\ A e. RR ) -> ( A =/= B <-> A =/= +oo ) )
45	42, 44	mpbird 166	. . . . . . 7 |- ( ( B = +oo /\ A e. RR ) -> A =/= B )
46	45	neneqd 2348	. . . . . 6 |- ( ( B = +oo /\ A e. RR ) -> -. A = B )
47		ltpnf 9687	. . . . . . . . 9 |- ( A e. RR -> A < +oo )
48	47	adantl 275	. . . . . . . 8 |- ( ( B = +oo /\ A e. RR ) -> A < +oo )
49		breq2 3969	. . . . . . . . 9 |- ( B = +oo -> ( A < B <-> A < +oo ) )
50	49	adantr 274	. . . . . . . 8 |- ( ( B = +oo /\ A e. RR ) -> ( A < B <-> A < +oo ) )
51	48, 50	mpbird 166	. . . . . . 7 |- ( ( B = +oo /\ A e. RR ) -> A < B )
52	51, 35, 36	3syl 17	. . . . . 6 |- ( ( B = +oo /\ A e. RR ) -> -. ( -. A < B /\ -. B < A ) )
53	46, 52	2falsed 692	. . . . 5 |- ( ( B = +oo /\ A e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
54		eqtr3 2177	. . . . . . 7 |- ( ( B = +oo /\ A = +oo ) -> B = A )
55	54	eqcomd 2163	. . . . . 6 |- ( ( B = +oo /\ A = +oo ) -> A = B )
56		pnfxr 7930	. . . . . . . . 9 |- +oo e. RR*
57		xrltnr 9686	. . . . . . . . 9 |- ( +oo e. RR* -> -. +oo < +oo )
58	56, 57	ax-mp 5	. . . . . . . 8 |- -. +oo < +oo
59		breq12 3970	. . . . . . . . 9 |- ( ( A = +oo /\ B = +oo ) -> ( A < B <-> +oo < +oo ) )
60	59	ancoms 266	. . . . . . . 8 |- ( ( B = +oo /\ A = +oo ) -> ( A < B <-> +oo < +oo ) )
61	58, 60	mtbiri 665	. . . . . . 7 |- ( ( B = +oo /\ A = +oo ) -> -. A < B )
62		breq12 3970	. . . . . . . 8 |- ( ( B = +oo /\ A = +oo ) -> ( B < A <-> +oo < +oo ) )
63	58, 62	mtbiri 665	. . . . . . 7 |- ( ( B = +oo /\ A = +oo ) -> -. B < A )
64	61, 63	jca 304	. . . . . 6 |- ( ( B = +oo /\ A = +oo ) -> ( -. A < B /\ -. B < A ) )
65	55, 64	2thd 174	. . . . 5 |- ( ( B = +oo /\ A = +oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
66		mnfnepnf 7933	. . . . . . . . 9 |- -oo =/= +oo
67		eqeq12 2170	. . . . . . . . . 10 |- ( ( A = -oo /\ B = +oo ) -> ( A = B <-> -oo = +oo ) )
68	67	necon3bid 2368	. . . . . . . . 9 |- ( ( A = -oo /\ B = +oo ) -> ( A =/= B <-> -oo =/= +oo ) )
69	66, 68	mpbiri 167	. . . . . . . 8 |- ( ( A = -oo /\ B = +oo ) -> A =/= B )
70	69	ancoms 266	. . . . . . 7 |- ( ( B = +oo /\ A = -oo ) -> A =/= B )
71	70	neneqd 2348	. . . . . 6 |- ( ( B = +oo /\ A = -oo ) -> -. A = B )
72		mnfltpnf 9692	. . . . . . . . 9 |- -oo < +oo
73		breq12 3970	. . . . . . . . 9 |- ( ( A = -oo /\ B = +oo ) -> ( A < B <-> -oo < +oo ) )
74	72, 73	mpbiri 167	. . . . . . . 8 |- ( ( A = -oo /\ B = +oo ) -> A < B )
75	74	ancoms 266	. . . . . . 7 |- ( ( B = +oo /\ A = -oo ) -> A < B )
76	75, 35, 36	3syl 17	. . . . . 6 |- ( ( B = +oo /\ A = -oo ) -> -. ( -. A < B /\ -. B < A ) )
77	71, 76	2falsed 692	. . . . 5 |- ( ( B = +oo /\ A = -oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
78	53, 65, 77	3jaodan 1288	. . . 4 |- ( ( B = +oo /\ ( A e. RR \/ A = +oo \/ A = -oo ) ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
79	78	ancoms 266	. . 3 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B = +oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
80		renemnf 7926	. . . . . . . . 9 |- ( A e. RR -> A =/= -oo )
81	80	adantl 275	. . . . . . . 8 |- ( ( B = -oo /\ A e. RR ) -> A =/= -oo )
82		neeq2 2341	. . . . . . . . 9 |- ( B = -oo -> ( A =/= B <-> A =/= -oo ) )
83	82	adantr 274	. . . . . . . 8 |- ( ( B = -oo /\ A e. RR ) -> ( A =/= B <-> A =/= -oo ) )
84	81, 83	mpbird 166	. . . . . . 7 |- ( ( B = -oo /\ A e. RR ) -> A =/= B )
85	84	neneqd 2348	. . . . . 6 |- ( ( B = -oo /\ A e. RR ) -> -. A = B )
86		mnflt 9690	. . . . . . . . 9 |- ( A e. RR -> -oo < A )
87	86	adantl 275	. . . . . . . 8 |- ( ( B = -oo /\ A e. RR ) -> -oo < A )
88		breq1 3968	. . . . . . . . 9 |- ( B = -oo -> ( B < A <-> -oo < A ) )
89	88	adantr 274	. . . . . . . 8 |- ( ( B = -oo /\ A e. RR ) -> ( B < A <-> -oo < A ) )
90	87, 89	mpbird 166	. . . . . . 7 |- ( ( B = -oo /\ A e. RR ) -> B < A )
91	90, 20	syl 14	. . . . . 6 |- ( ( B = -oo /\ A e. RR ) -> -. ( -. A < B /\ -. B < A ) )
92	85, 91	2falsed 692	. . . . 5 |- ( ( B = -oo /\ A e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
93	66	neii 2329	. . . . . . . . . 10 |- -. -oo = +oo
94		eqeq12 2170	. . . . . . . . . 10 |- ( ( B = -oo /\ A = +oo ) -> ( B = A <-> -oo = +oo ) )
95	93, 94	mtbiri 665	. . . . . . . . 9 |- ( ( B = -oo /\ A = +oo ) -> -. B = A )
96	95	neneqad 2406	. . . . . . . 8 |- ( ( B = -oo /\ A = +oo ) -> B =/= A )
97	96	necomd 2413	. . . . . . 7 |- ( ( B = -oo /\ A = +oo ) -> A =/= B )
98	97	neneqd 2348	. . . . . 6 |- ( ( B = -oo /\ A = +oo ) -> -. A = B )
99		breq12 3970	. . . . . . . 8 |- ( ( B = -oo /\ A = +oo ) -> ( B < A <-> -oo < +oo ) )
100	72, 99	mpbiri 167	. . . . . . 7 |- ( ( B = -oo /\ A = +oo ) -> B < A )
101	100, 20	syl 14	. . . . . 6 |- ( ( B = -oo /\ A = +oo ) -> -. ( -. A < B /\ -. B < A ) )
102	98, 101	2falsed 692	. . . . 5 |- ( ( B = -oo /\ A = +oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
103		eqtr3 2177	. . . . . . 7 |- ( ( A = -oo /\ B = -oo ) -> A = B )
104	103	ancoms 266	. . . . . 6 |- ( ( B = -oo /\ A = -oo ) -> A = B )
105		mnfxr 7934	. . . . . . . . 9 |- -oo e. RR*
106		xrltnr 9686	. . . . . . . . 9 |- ( -oo e. RR* -> -. -oo < -oo )
107	105, 106	ax-mp 5	. . . . . . . 8 |- -. -oo < -oo
108		breq12 3970	. . . . . . . . 9 |- ( ( A = -oo /\ B = -oo ) -> ( A < B <-> -oo < -oo ) )
109	108	ancoms 266	. . . . . . . 8 |- ( ( B = -oo /\ A = -oo ) -> ( A < B <-> -oo < -oo ) )
110	107, 109	mtbiri 665	. . . . . . 7 |- ( ( B = -oo /\ A = -oo ) -> -. A < B )
111		breq12 3970	. . . . . . . 8 |- ( ( B = -oo /\ A = -oo ) -> ( B < A <-> -oo < -oo ) )
112	107, 111	mtbiri 665	. . . . . . 7 |- ( ( B = -oo /\ A = -oo ) -> -. B < A )
113	110, 112	jca 304	. . . . . 6 |- ( ( B = -oo /\ A = -oo ) -> ( -. A < B /\ -. B < A ) )
114	104, 113	2thd 174	. . . . 5 |- ( ( B = -oo /\ A = -oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
115	92, 102, 114	3jaodan 1288	. . . 4 |- ( ( B = -oo /\ ( A e. RR \/ A = +oo \/ A = -oo ) ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
116	115	ancoms 266	. . 3 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B = -oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
117	40, 79, 116	3jaodan 1288	. 2 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
118	1, 2, 117	syl2anb 289	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   \/ w3o 962   = wceq 1335   e. wcel 2128   =/= wne 2327   class class class wbr 3965   RRcr 7731   +oocpnf 7909   -oocmnf 7910   RR*cxr 7911   < clt 7912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-pre-ltirr 7844  ax-pre-apti 7847

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4592  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917

This theorem is referenced by:  xrletri3  9709  iccid  9829  xrmaxleim  11141  xrmaxif  11148  xrmaxaddlem  11157  infxrnegsupex  11160  bdxmet  12912