Theorem xrlttri3 8872

Description: Extended real version of lttri3 7191. (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 8850	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 8850	. 2 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		lttri3 7191	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
4	3	ancoms 264	. . . . 5 |- ( ( B e. RR /\ A e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
5		renepnf 7166	. . . . . . . . . 10 |- ( B e. RR -> B =/= +oo )
6	5	adantr 270	. . . . . . . . 9 |- ( ( B e. RR /\ A = +oo ) -> B =/= +oo )
7		neeq2 2259	. . . . . . . . . 10 |- ( A = +oo -> ( B =/= A <-> B =/= +oo ) )
8	7	adantl 271	. . . . . . . . 9 |- ( ( B e. RR /\ A = +oo ) -> ( B =/= A <-> B =/= +oo ) )
9	6, 8	mpbird 165	. . . . . . . 8 |- ( ( B e. RR /\ A = +oo ) -> B =/= A )
10	9	necomd 2331	. . . . . . 7 |- ( ( B e. RR /\ A = +oo ) -> A =/= B )
11	10	neneqd 2266	. . . . . 6 |- ( ( B e. RR /\ A = +oo ) -> -. A = B )
12		ltpnf 8856	. . . . . . . . 9 |- ( B e. RR -> B < +oo )
13	12	adantr 270	. . . . . . . 8 |- ( ( B e. RR /\ A = +oo ) -> B < +oo )
14		breq2 3789	. . . . . . . . 9 |- ( A = +oo -> ( B < A <-> B < +oo ) )
15	14	adantl 271	. . . . . . . 8 |- ( ( B e. RR /\ A = +oo ) -> ( B < A <-> B < +oo ) )
16	13, 15	mpbird 165	. . . . . . 7 |- ( ( B e. RR /\ A = +oo ) -> B < A )
17		notnot 591	. . . . . . . . 9 |- ( ( A < B \/ B < A ) -> -. -. ( A < B \/ B < A ) )
18	17	olcs 687	. . . . . . . 8 |- ( B < A -> -. -. ( A < B \/ B < A ) )
19		ioran 701	. . . . . . . 8 |- ( -. ( A < B \/ B < A ) <-> ( -. A < B /\ -. B < A ) )
20	18, 19	sylnib 633	. . . . . . 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 650	. . . . 5 |- ( ( B e. RR /\ A = +oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
23		renemnf 7167	. . . . . . . . . 10 |- ( B e. RR -> B =/= -oo )
24	23	adantr 270	. . . . . . . . 9 |- ( ( B e. RR /\ A = -oo ) -> B =/= -oo )
25		neeq2 2259	. . . . . . . . . 10 |- ( A = -oo -> ( B =/= A <-> B =/= -oo ) )
26	25	adantl 271	. . . . . . . . 9 |- ( ( B e. RR /\ A = -oo ) -> ( B =/= A <-> B =/= -oo ) )
27	24, 26	mpbird 165	. . . . . . . 8 |- ( ( B e. RR /\ A = -oo ) -> B =/= A )
28	27	necomd 2331	. . . . . . 7 |- ( ( B e. RR /\ A = -oo ) -> A =/= B )
29	28	neneqd 2266	. . . . . 6 |- ( ( B e. RR /\ A = -oo ) -> -. A = B )
30		mnflt 8858	. . . . . . . . 9 |- ( B e. RR -> -oo < B )
31	30	adantr 270	. . . . . . . 8 |- ( ( B e. RR /\ A = -oo ) -> -oo < B )
32		breq1 3788	. . . . . . . . 9 |- ( A = -oo -> ( A < B <-> -oo < B ) )
33	32	adantl 271	. . . . . . . 8 |- ( ( B e. RR /\ A = -oo ) -> ( A < B <-> -oo < B ) )
34	31, 33	mpbird 165	. . . . . . 7 |- ( ( B e. RR /\ A = -oo ) -> A < B )
35		orc 665	. . . . . . 7 |- ( A < B -> ( A < B \/ B < A ) )
36		oranim 840	. . . . . . 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 650	. . . . 5 |- ( ( B e. RR /\ A = -oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
39	4, 22, 38	3jaodan 1237	. . . 4 |- ( ( B e. RR /\ ( A e. RR \/ A = +oo \/ A = -oo ) ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
40	39	ancoms 264	. . 3 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
41		renepnf 7166	. . . . . . . . 9 |- ( A e. RR -> A =/= +oo )
42	41	adantl 271	. . . . . . . 8 |- ( ( B = +oo /\ A e. RR ) -> A =/= +oo )
43		neeq2 2259	. . . . . . . . 9 |- ( B = +oo -> ( A =/= B <-> A =/= +oo ) )
44	43	adantr 270	. . . . . . . 8 |- ( ( B = +oo /\ A e. RR ) -> ( A =/= B <-> A =/= +oo ) )
45	42, 44	mpbird 165	. . . . . . 7 |- ( ( B = +oo /\ A e. RR ) -> A =/= B )
46	45	neneqd 2266	. . . . . 6 |- ( ( B = +oo /\ A e. RR ) -> -. A = B )
47		ltpnf 8856	. . . . . . . . 9 |- ( A e. RR -> A < +oo )
48	47	adantl 271	. . . . . . . 8 |- ( ( B = +oo /\ A e. RR ) -> A < +oo )
49		breq2 3789	. . . . . . . . 9 |- ( B = +oo -> ( A < B <-> A < +oo ) )
50	49	adantr 270	. . . . . . . 8 |- ( ( B = +oo /\ A e. RR ) -> ( A < B <-> A < +oo ) )
51	48, 50	mpbird 165	. . . . . . 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 650	. . . . 5 |- ( ( B = +oo /\ A e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
54		eqtr3 2100	. . . . . . 7 |- ( ( B = +oo /\ A = +oo ) -> B = A )
55	54	eqcomd 2086	. . . . . 6 |- ( ( B = +oo /\ A = +oo ) -> A = B )
56		pnfxr 8846	. . . . . . . . 9 |- +oo e. RR*
57		xrltnr 8855	. . . . . . . . 9 |- ( +oo e. RR* -> -. +oo < +oo )
58	56, 57	ax-mp 7	. . . . . . . 8 |- -. +oo < +oo
59		breq12 3790	. . . . . . . . 9 |- ( ( A = +oo /\ B = +oo ) -> ( A < B <-> +oo < +oo ) )
60	59	ancoms 264	. . . . . . . 8 |- ( ( B = +oo /\ A = +oo ) -> ( A < B <-> +oo < +oo ) )
61	58, 60	mtbiri 632	. . . . . . 7 |- ( ( B = +oo /\ A = +oo ) -> -. A < B )
62		breq12 3790	. . . . . . . 8 |- ( ( B = +oo /\ A = +oo ) -> ( B < A <-> +oo < +oo ) )
63	58, 62	mtbiri 632	. . . . . . 7 |- ( ( B = +oo /\ A = +oo ) -> -. B < A )
64	61, 63	jca 300	. . . . . 6 |- ( ( B = +oo /\ A = +oo ) -> ( -. A < B /\ -. B < A ) )
65	55, 64	2thd 173	. . . . 5 |- ( ( B = +oo /\ A = +oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
66		mnfnepnf 8852	. . . . . . . . 9 |- -oo =/= +oo
67		eqeq12 2093	. . . . . . . . . 10 |- ( ( A = -oo /\ B = +oo ) -> ( A = B <-> -oo = +oo ) )
68	67	necon3bid 2286	. . . . . . . . 9 |- ( ( A = -oo /\ B = +oo ) -> ( A =/= B <-> -oo =/= +oo ) )
69	66, 68	mpbiri 166	. . . . . . . 8 |- ( ( A = -oo /\ B = +oo ) -> A =/= B )
70	69	ancoms 264	. . . . . . 7 |- ( ( B = +oo /\ A = -oo ) -> A =/= B )
71	70	neneqd 2266	. . . . . 6 |- ( ( B = +oo /\ A = -oo ) -> -. A = B )
72		mnfltpnf 8860	. . . . . . . . 9 |- -oo < +oo
73		breq12 3790	. . . . . . . . 9 |- ( ( A = -oo /\ B = +oo ) -> ( A < B <-> -oo < +oo ) )
74	72, 73	mpbiri 166	. . . . . . . 8 |- ( ( A = -oo /\ B = +oo ) -> A < B )
75	74	ancoms 264	. . . . . . 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 650	. . . . 5 |- ( ( B = +oo /\ A = -oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
78	53, 65, 77	3jaodan 1237	. . . 4 |- ( ( B = +oo /\ ( A e. RR \/ A = +oo \/ A = -oo ) ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
79	78	ancoms 264	. . 3 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B = +oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
80		renemnf 7167	. . . . . . . . 9 |- ( A e. RR -> A =/= -oo )
81	80	adantl 271	. . . . . . . 8 |- ( ( B = -oo /\ A e. RR ) -> A =/= -oo )
82		neeq2 2259	. . . . . . . . 9 |- ( B = -oo -> ( A =/= B <-> A =/= -oo ) )
83	82	adantr 270	. . . . . . . 8 |- ( ( B = -oo /\ A e. RR ) -> ( A =/= B <-> A =/= -oo ) )
84	81, 83	mpbird 165	. . . . . . 7 |- ( ( B = -oo /\ A e. RR ) -> A =/= B )
85	84	neneqd 2266	. . . . . 6 |- ( ( B = -oo /\ A e. RR ) -> -. A = B )
86		mnflt 8858	. . . . . . . . 9 |- ( A e. RR -> -oo < A )
87	86	adantl 271	. . . . . . . 8 |- ( ( B = -oo /\ A e. RR ) -> -oo < A )
88		breq1 3788	. . . . . . . . 9 |- ( B = -oo -> ( B < A <-> -oo < A ) )
89	88	adantr 270	. . . . . . . 8 |- ( ( B = -oo /\ A e. RR ) -> ( B < A <-> -oo < A ) )
90	87, 89	mpbird 165	. . . . . . 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 650	. . . . 5 |- ( ( B = -oo /\ A e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
93	66	neii 2247	. . . . . . . . . 10 |- -. -oo = +oo
94		eqeq12 2093	. . . . . . . . . 10 |- ( ( B = -oo /\ A = +oo ) -> ( B = A <-> -oo = +oo ) )
95	93, 94	mtbiri 632	. . . . . . . . 9 |- ( ( B = -oo /\ A = +oo ) -> -. B = A )
96	95	neneqad 2324	. . . . . . . 8 |- ( ( B = -oo /\ A = +oo ) -> B =/= A )
97	96	necomd 2331	. . . . . . 7 |- ( ( B = -oo /\ A = +oo ) -> A =/= B )
98	97	neneqd 2266	. . . . . 6 |- ( ( B = -oo /\ A = +oo ) -> -. A = B )
99		breq12 3790	. . . . . . . 8 |- ( ( B = -oo /\ A = +oo ) -> ( B < A <-> -oo < +oo ) )
100	72, 99	mpbiri 166	. . . . . . 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 650	. . . . 5 |- ( ( B = -oo /\ A = +oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
103		eqtr3 2100	. . . . . . 7 |- ( ( A = -oo /\ B = -oo ) -> A = B )
104	103	ancoms 264	. . . . . 6 |- ( ( B = -oo /\ A = -oo ) -> A = B )
105		mnfxr 8848	. . . . . . . . 9 |- -oo e. RR*
106		xrltnr 8855	. . . . . . . . 9 |- ( -oo e. RR* -> -. -oo < -oo )
107	105, 106	ax-mp 7	. . . . . . . 8 |- -. -oo < -oo
108		breq12 3790	. . . . . . . . 9 |- ( ( A = -oo /\ B = -oo ) -> ( A < B <-> -oo < -oo ) )
109	108	ancoms 264	. . . . . . . 8 |- ( ( B = -oo /\ A = -oo ) -> ( A < B <-> -oo < -oo ) )
110	107, 109	mtbiri 632	. . . . . . 7 |- ( ( B = -oo /\ A = -oo ) -> -. A < B )
111		breq12 3790	. . . . . . . 8 |- ( ( B = -oo /\ A = -oo ) -> ( B < A <-> -oo < -oo ) )
112	107, 111	mtbiri 632	. . . . . . 7 |- ( ( B = -oo /\ A = -oo ) -> -. B < A )
113	110, 112	jca 300	. . . . . 6 |- ( ( B = -oo /\ A = -oo ) -> ( -. A < B /\ -. B < A ) )
114	104, 113	2thd 173	. . . . 5 |- ( ( B = -oo /\ A = -oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
115	92, 102, 114	3jaodan 1237	. . . 4 |- ( ( B = -oo /\ ( A e. RR \/ A = +oo \/ A = -oo ) ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
116	115	ancoms 264	. . 3 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B = -oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
117	40, 79, 116	3jaodan 1237	. 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 285	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 661   \/ w3o 918   = wceq 1284   e. wcel 1433   =/= wne 2245   class class class wbr 3785   RRcr 6980   +oocpnf 7150   -oocmnf 7151   RR*cxr 7152   < clt 7153

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-pre-ltirr 7088  ax-pre-apti 7091

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-xp 4369  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158

This theorem is referenced by:  xrletri3  8875  iccid  8948