Theorem xrlttri3 9993

Description: Extended real version of lttri3 8226. (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 9972	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 9972	. 2 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		lttri3 8226	. . . . . 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 8194	. . . . . . . . . 10 |- ( B e. RR -> B =/= +oo )
6	5	adantr 276	. . . . . . . . 9 |- ( ( B e. RR /\ A = +oo ) -> B =/= +oo )
7		neeq2 2414	. . . . . . . . . 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 2486	. . . . . . 7 |- ( ( B e. RR /\ A = +oo ) -> A =/= B )
11	10	neneqd 2421	. . . . . 6 |- ( ( B e. RR /\ A = +oo ) -> -. A = B )
12		ltpnf 9976	. . . . . . . . 9 |- ( B e. RR -> B < +oo )
13	12	adantr 276	. . . . . . . 8 |- ( ( B e. RR /\ A = +oo ) -> B < +oo )
14		breq2 4087	. . . . . . . . 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 632	. . . . . . . . 9 |- ( ( A < B \/ B < A ) -> -. -. ( A < B \/ B < A ) )
18	17	olcs 741	. . . . . . . 8 |- ( B < A -> -. -. ( A < B \/ B < A ) )
19		ioran 757	. . . . . . . 8 |- ( -. ( A < B \/ B < A ) <-> ( -. A < B /\ -. B < A ) )
20	18, 19	sylnib 680	. . . . . . 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 707	. . . . 5 |- ( ( B e. RR /\ A = +oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
23		renemnf 8195	. . . . . . . . . 10 |- ( B e. RR -> B =/= -oo )
24	23	adantr 276	. . . . . . . . 9 |- ( ( B e. RR /\ A = -oo ) -> B =/= -oo )
25		neeq2 2414	. . . . . . . . . 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 2486	. . . . . . 7 |- ( ( B e. RR /\ A = -oo ) -> A =/= B )
29	28	neneqd 2421	. . . . . 6 |- ( ( B e. RR /\ A = -oo ) -> -. A = B )
30		mnflt 9979	. . . . . . . . 9 |- ( B e. RR -> -oo < B )
31	30	adantr 276	. . . . . . . 8 |- ( ( B e. RR /\ A = -oo ) -> -oo < B )
32		breq1 4086	. . . . . . . . 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 717	. . . . . . 7 |- ( A < B -> ( A < B \/ B < A ) )
36		oranim 786	. . . . . . 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 707	. . . . 5 |- ( ( B e. RR /\ A = -oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
39	4, 22, 38	3jaodan 1340	. . . 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 8194	. . . . . . . . 9 |- ( A e. RR -> A =/= +oo )
42	41	adantl 277	. . . . . . . 8 |- ( ( B = +oo /\ A e. RR ) -> A =/= +oo )
43		neeq2 2414	. . . . . . . . 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 2421	. . . . . 6 |- ( ( B = +oo /\ A e. RR ) -> -. A = B )
47		ltpnf 9976	. . . . . . . . 9 |- ( A e. RR -> A < +oo )
48	47	adantl 277	. . . . . . . 8 |- ( ( B = +oo /\ A e. RR ) -> A < +oo )
49		breq2 4087	. . . . . . . . 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 707	. . . . 5 |- ( ( B = +oo /\ A e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
54		eqtr3 2249	. . . . . . 7 |- ( ( B = +oo /\ A = +oo ) -> B = A )
55	54	eqcomd 2235	. . . . . 6 |- ( ( B = +oo /\ A = +oo ) -> A = B )
56		pnfxr 8199	. . . . . . . . 9 |- +oo e. RR*
57		xrltnr 9975	. . . . . . . . 9 |- ( +oo e. RR* -> -. +oo < +oo )
58	56, 57	ax-mp 5	. . . . . . . 8 |- -. +oo < +oo
59		breq12 4088	. . . . . . . . 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 679	. . . . . . 7 |- ( ( B = +oo /\ A = +oo ) -> -. A < B )
62		breq12 4088	. . . . . . . 8 |- ( ( B = +oo /\ A = +oo ) -> ( B < A <-> +oo < +oo ) )
63	58, 62	mtbiri 679	. . . . . . 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 8202	. . . . . . . . 9 |- -oo =/= +oo
67		eqeq12 2242	. . . . . . . . . 10 |- ( ( A = -oo /\ B = +oo ) -> ( A = B <-> -oo = +oo ) )
68	67	necon3bid 2441	. . . . . . . . 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 2421	. . . . . 6 |- ( ( B = +oo /\ A = -oo ) -> -. A = B )
72		mnfltpnf 9981	. . . . . . . . 9 |- -oo < +oo
73		breq12 4088	. . . . . . . . 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 707	. . . . 5 |- ( ( B = +oo /\ A = -oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
78	53, 65, 77	3jaodan 1340	. . . 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 8195	. . . . . . . . 9 |- ( A e. RR -> A =/= -oo )
81	80	adantl 277	. . . . . . . 8 |- ( ( B = -oo /\ A e. RR ) -> A =/= -oo )
82		neeq2 2414	. . . . . . . . 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 2421	. . . . . 6 |- ( ( B = -oo /\ A e. RR ) -> -. A = B )
86		mnflt 9979	. . . . . . . . 9 |- ( A e. RR -> -oo < A )
87	86	adantl 277	. . . . . . . 8 |- ( ( B = -oo /\ A e. RR ) -> -oo < A )
88		breq1 4086	. . . . . . . . 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 707	. . . . 5 |- ( ( B = -oo /\ A e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
93	66	neii 2402	. . . . . . . . . 10 |- -. -oo = +oo
94		eqeq12 2242	. . . . . . . . . 10 |- ( ( B = -oo /\ A = +oo ) -> ( B = A <-> -oo = +oo ) )
95	93, 94	mtbiri 679	. . . . . . . . 9 |- ( ( B = -oo /\ A = +oo ) -> -. B = A )
96	95	neneqad 2479	. . . . . . . 8 |- ( ( B = -oo /\ A = +oo ) -> B =/= A )
97	96	necomd 2486	. . . . . . 7 |- ( ( B = -oo /\ A = +oo ) -> A =/= B )
98	97	neneqd 2421	. . . . . 6 |- ( ( B = -oo /\ A = +oo ) -> -. A = B )
99		breq12 4088	. . . . . . . 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 707	. . . . 5 |- ( ( B = -oo /\ A = +oo ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
103		eqtr3 2249	. . . . . . 7 |- ( ( A = -oo /\ B = -oo ) -> A = B )
104	103	ancoms 268	. . . . . 6 |- ( ( B = -oo /\ A = -oo ) -> A = B )
105		mnfxr 8203	. . . . . . . . 9 |- -oo e. RR*
106		xrltnr 9975	. . . . . . . . 9 |- ( -oo e. RR* -> -. -oo < -oo )
107	105, 106	ax-mp 5	. . . . . . . 8 |- -. -oo < -oo
108		breq12 4088	. . . . . . . . 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 679	. . . . . . 7 |- ( ( B = -oo /\ A = -oo ) -> -. A < B )
111		breq12 4088	. . . . . . . 8 |- ( ( B = -oo /\ A = -oo ) -> ( B < A <-> -oo < -oo ) )
112	107, 111	mtbiri 679	. . . . . . 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 1340	. . . 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 1340	. 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 713   \/ w3o 1001   = wceq 1395   e. wcel 2200   =/= wne 2400   class class class wbr 4083   RRcr 7998   +oocpnf 8178   -oocmnf 8179   RR*cxr 8180   < clt 8181

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-pre-ltirr 8111  ax-pre-apti 8114

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186

This theorem is referenced by:  xrletri3  10000  iccid  10121  xrmaxleim  11755  xrmaxif  11762  xrmaxaddlem  11771  infxrnegsupex  11774  bdxmet  15175