Theorem xrlttri3 9889

Description: Extended real version of lttri3 8123. (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 9868	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 9868	. 2 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		lttri3 8123	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
4	3	ancoms 268	. . . . 5 |- ( ( B e. RR /\ A e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
5		renepnf 8091	. . . . . . . . . 10 |- ( B e. RR -> B =/= +oo )
6	5	adantr 276	. . . . . . . . 9 |- ( ( B e. RR /\ A = +oo ) -> B =/= +oo )
7		neeq2 2381	. . . . . . . . . 10 |- ( A = +oo -> ( B =/= A <-> B =/= +oo ) )
8	7	adantl 277	. . . . . . . . 9 |- ( ( B e. RR /\ A = +oo ) -> ( B =/= A <-> B =/= +oo ) )
9	6, 8	mpbird 167	. . . . . . . 8 |- ( ( B e. RR /\ A = +oo ) -> B =/= A )
10	9	necomd 2453	. . . . . . 7 |- ( ( B e. RR /\ A = +oo ) -> A =/= B )
11	10	neneqd 2388	. . . . . 6 |- ( ( B e. RR /\ A = +oo ) -> -. A = B )
12		ltpnf 9872	. . . . . . . . 9 |- ( B e. RR -> B < +oo )
13	12	adantr 276	. . . . . . . 8 |- ( ( B e. RR /\ A = +oo ) -> B < +oo )
14		breq2 4038	. . . . . . . . 9 |- ( A = +oo -> ( B < A <-> B < +oo ) )
15	14	adantl 277	. . . . . . . 8 |- ( ( B e. RR /\ A = +oo ) -> ( B < A <-> B < +oo ) )
16	13, 15	mpbird 167	. . . . . . 7 |- ( ( B e. RR /\ A = +oo ) -> B < A )
17		notnot 630	. . . . . . . . 9 |- ( ( A < B \/ B < A ) -> -. -. ( A < B \/ B < A ) )
18	17	olcs 737	. . . . . . . 8 |- ( B < A -> -. -. ( A < B \/ B < A ) )
19		ioran 753	. . . . . . . 8 |- ( -. ( A < B \/ B < A ) <-> ( -. A < B /\ -. B < A ) )
20	18, 19	sylnib 677	. . . . . . 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 703	. . . . 5 |- ( ( B e. RR /\ A = +oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
23		renemnf 8092	. . . . . . . . . 10 |- ( B e. RR -> B =/= -oo )
24	23	adantr 276	. . . . . . . . 9 |- ( ( B e. RR /\ A = -oo ) -> B =/= -oo )
25		neeq2 2381	. . . . . . . . . 10 |- ( A = -oo -> ( B =/= A <-> B =/= -oo ) )
26	25	adantl 277	. . . . . . . . 9 |- ( ( B e. RR /\ A = -oo ) -> ( B =/= A <-> B =/= -oo ) )
27	24, 26	mpbird 167	. . . . . . . 8 |- ( ( B e. RR /\ A = -oo ) -> B =/= A )
28	27	necomd 2453	. . . . . . 7 |- ( ( B e. RR /\ A = -oo ) -> A =/= B )
29	28	neneqd 2388	. . . . . 6 |- ( ( B e. RR /\ A = -oo ) -> -. A = B )
30		mnflt 9875	. . . . . . . . 9 |- ( B e. RR -> -oo < B )
31	30	adantr 276	. . . . . . . 8 |- ( ( B e. RR /\ A = -oo ) -> -oo < B )
32		breq1 4037	. . . . . . . . 9 |- ( A = -oo -> ( A < B <-> -oo < B ) )
33	32	adantl 277	. . . . . . . 8 |- ( ( B e. RR /\ A = -oo ) -> ( A < B <-> -oo < B ) )
34	31, 33	mpbird 167	. . . . . . 7 |- ( ( B e. RR /\ A = -oo ) -> A < B )
35		orc 713	. . . . . . 7 |- ( A < B -> ( A < B \/ B < A ) )
36		oranim 782	. . . . . . 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 703	. . . . 5 |- ( ( B e. RR /\ A = -oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
39	4, 22, 38	3jaodan 1317	. . . 4 |- ( ( B e. RR /\ ( A e. RR \/ A = +oo \/ A = -oo ) ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
40	39	ancoms 268	. . 3 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
41		renepnf 8091	. . . . . . . . 9 |- ( A e. RR -> A =/= +oo )
42	41	adantl 277	. . . . . . . 8 |- ( ( B = +oo /\ A e. RR ) -> A =/= +oo )
43		neeq2 2381	. . . . . . . . 9 |- ( B = +oo -> ( A =/= B <-> A =/= +oo ) )
44	43	adantr 276	. . . . . . . 8 |- ( ( B = +oo /\ A e. RR ) -> ( A =/= B <-> A =/= +oo ) )
45	42, 44	mpbird 167	. . . . . . 7 |- ( ( B = +oo /\ A e. RR ) -> A =/= B )
46	45	neneqd 2388	. . . . . 6 |- ( ( B = +oo /\ A e. RR ) -> -. A = B )
47		ltpnf 9872	. . . . . . . . 9 |- ( A e. RR -> A < +oo )
48	47	adantl 277	. . . . . . . 8 |- ( ( B = +oo /\ A e. RR ) -> A < +oo )
49		breq2 4038	. . . . . . . . 9 |- ( B = +oo -> ( A < B <-> A < +oo ) )
50	49	adantr 276	. . . . . . . 8 |- ( ( B = +oo /\ A e. RR ) -> ( A < B <-> A < +oo ) )
51	48, 50	mpbird 167	. . . . . . 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 703	. . . . 5 |- ( ( B = +oo /\ A e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
54		eqtr3 2216	. . . . . . 7 |- ( ( B = +oo /\ A = +oo ) -> B = A )
55	54	eqcomd 2202	. . . . . 6 |- ( ( B = +oo /\ A = +oo ) -> A = B )
56		pnfxr 8096	. . . . . . . . 9 |- +oo e. RR*
57		xrltnr 9871	. . . . . . . . 9 |- ( +oo e. RR* -> -. +oo < +oo )
58	56, 57	ax-mp 5	. . . . . . . 8 |- -. +oo < +oo
59		breq12 4039	. . . . . . . . 9 |- ( ( A = +oo /\ B = +oo ) -> ( A < B <-> +oo < +oo ) )
60	59	ancoms 268	. . . . . . . 8 |- ( ( B = +oo /\ A = +oo ) -> ( A < B <-> +oo < +oo ) )
61	58, 60	mtbiri 676	. . . . . . 7 |- ( ( B = +oo /\ A = +oo ) -> -. A < B )
62		breq12 4039	. . . . . . . 8 |- ( ( B = +oo /\ A = +oo ) -> ( B < A <-> +oo < +oo ) )
63	58, 62	mtbiri 676	. . . . . . 7 |- ( ( B = +oo /\ A = +oo ) -> -. B < A )
64	61, 63	jca 306	. . . . . 6 |- ( ( B = +oo /\ A = +oo ) -> ( -. A < B /\ -. B < A ) )
65	55, 64	2thd 175	. . . . 5 |- ( ( B = +oo /\ A = +oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
66		mnfnepnf 8099	. . . . . . . . 9 |- -oo =/= +oo
67		eqeq12 2209	. . . . . . . . . 10 |- ( ( A = -oo /\ B = +oo ) -> ( A = B <-> -oo = +oo ) )
68	67	necon3bid 2408	. . . . . . . . 9 |- ( ( A = -oo /\ B = +oo ) -> ( A =/= B <-> -oo =/= +oo ) )
69	66, 68	mpbiri 168	. . . . . . . 8 |- ( ( A = -oo /\ B = +oo ) -> A =/= B )
70	69	ancoms 268	. . . . . . 7 |- ( ( B = +oo /\ A = -oo ) -> A =/= B )
71	70	neneqd 2388	. . . . . 6 |- ( ( B = +oo /\ A = -oo ) -> -. A = B )
72		mnfltpnf 9877	. . . . . . . . 9 |- -oo < +oo
73		breq12 4039	. . . . . . . . 9 |- ( ( A = -oo /\ B = +oo ) -> ( A < B <-> -oo < +oo ) )
74	72, 73	mpbiri 168	. . . . . . . 8 |- ( ( A = -oo /\ B = +oo ) -> A < B )
75	74	ancoms 268	. . . . . . 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 703	. . . . 5 |- ( ( B = +oo /\ A = -oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
78	53, 65, 77	3jaodan 1317	. . . 4 |- ( ( B = +oo /\ ( A e. RR \/ A = +oo \/ A = -oo ) ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
79	78	ancoms 268	. . 3 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B = +oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
80		renemnf 8092	. . . . . . . . 9 |- ( A e. RR -> A =/= -oo )
81	80	adantl 277	. . . . . . . 8 |- ( ( B = -oo /\ A e. RR ) -> A =/= -oo )
82		neeq2 2381	. . . . . . . . 9 |- ( B = -oo -> ( A =/= B <-> A =/= -oo ) )
83	82	adantr 276	. . . . . . . 8 |- ( ( B = -oo /\ A e. RR ) -> ( A =/= B <-> A =/= -oo ) )
84	81, 83	mpbird 167	. . . . . . 7 |- ( ( B = -oo /\ A e. RR ) -> A =/= B )
85	84	neneqd 2388	. . . . . 6 |- ( ( B = -oo /\ A e. RR ) -> -. A = B )
86		mnflt 9875	. . . . . . . . 9 |- ( A e. RR -> -oo < A )
87	86	adantl 277	. . . . . . . 8 |- ( ( B = -oo /\ A e. RR ) -> -oo < A )
88		breq1 4037	. . . . . . . . 9 |- ( B = -oo -> ( B < A <-> -oo < A ) )
89	88	adantr 276	. . . . . . . 8 |- ( ( B = -oo /\ A e. RR ) -> ( B < A <-> -oo < A ) )
90	87, 89	mpbird 167	. . . . . . 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 703	. . . . 5 |- ( ( B = -oo /\ A e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
93	66	neii 2369	. . . . . . . . . 10 |- -. -oo = +oo
94		eqeq12 2209	. . . . . . . . . 10 |- ( ( B = -oo /\ A = +oo ) -> ( B = A <-> -oo = +oo ) )
95	93, 94	mtbiri 676	. . . . . . . . 9 |- ( ( B = -oo /\ A = +oo ) -> -. B = A )
96	95	neneqad 2446	. . . . . . . 8 |- ( ( B = -oo /\ A = +oo ) -> B =/= A )
97	96	necomd 2453	. . . . . . 7 |- ( ( B = -oo /\ A = +oo ) -> A =/= B )
98	97	neneqd 2388	. . . . . 6 |- ( ( B = -oo /\ A = +oo ) -> -. A = B )
99		breq12 4039	. . . . . . . 8 |- ( ( B = -oo /\ A = +oo ) -> ( B < A <-> -oo < +oo ) )
100	72, 99	mpbiri 168	. . . . . . 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 703	. . . . 5 |- ( ( B = -oo /\ A = +oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
103		eqtr3 2216	. . . . . . 7 |- ( ( A = -oo /\ B = -oo ) -> A = B )
104	103	ancoms 268	. . . . . 6 |- ( ( B = -oo /\ A = -oo ) -> A = B )
105		mnfxr 8100	. . . . . . . . 9 |- -oo e. RR*
106		xrltnr 9871	. . . . . . . . 9 |- ( -oo e. RR* -> -. -oo < -oo )
107	105, 106	ax-mp 5	. . . . . . . 8 |- -. -oo < -oo
108		breq12 4039	. . . . . . . . 9 |- ( ( A = -oo /\ B = -oo ) -> ( A < B <-> -oo < -oo ) )
109	108	ancoms 268	. . . . . . . 8 |- ( ( B = -oo /\ A = -oo ) -> ( A < B <-> -oo < -oo ) )
110	107, 109	mtbiri 676	. . . . . . 7 |- ( ( B = -oo /\ A = -oo ) -> -. A < B )
111		breq12 4039	. . . . . . . 8 |- ( ( B = -oo /\ A = -oo ) -> ( B < A <-> -oo < -oo ) )
112	107, 111	mtbiri 676	. . . . . . 7 |- ( ( B = -oo /\ A = -oo ) -> -. B < A )
113	110, 112	jca 306	. . . . . 6 |- ( ( B = -oo /\ A = -oo ) -> ( -. A < B /\ -. B < A ) )
114	104, 113	2thd 175	. . . . 5 |- ( ( B = -oo /\ A = -oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
115	92, 102, 114	3jaodan 1317	. . . 4 |- ( ( B = -oo /\ ( A e. RR \/ A = +oo \/ A = -oo ) ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
116	115	ancoms 268	. . 3 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B = -oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
117	40, 79, 116	3jaodan 1317	. 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 291	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   \/ w3o 979   = wceq 1364   e. wcel 2167   =/= wne 2367   class class class wbr 4034   RRcr 7895   +oocpnf 8075   -oocmnf 8076   RR*cxr 8077   < clt 8078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-pre-ltirr 8008  ax-pre-apti 8011

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-xp 4670  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083

This theorem is referenced by:  xrletri3  9896  iccid  10017  xrmaxleim  11426  xrmaxif  11433  xrmaxaddlem  11442  infxrnegsupex  11445  bdxmet  14821