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Mirrors > Home > ILE Home > Th. List > xrlttri3 | Unicode version |
Description: Extended real version of lttri3 7098. (Contributed by NM, 9-Feb-2006.) |
Ref | Expression |
---|---|
xrlttri3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 8696 | . 2 | |
2 | elxr 8696 | . 2 | |
3 | lttri3 7098 | . . . . . 6 | |
4 | 3 | ancoms 255 | . . . . 5 |
5 | renepnf 7073 | . . . . . . . . . 10 | |
6 | 5 | adantr 261 | . . . . . . . . 9 |
7 | neeq2 2219 | . . . . . . . . . 10 | |
8 | 7 | adantl 262 | . . . . . . . . 9 |
9 | 6, 8 | mpbird 156 | . . . . . . . 8 |
10 | 9 | necomd 2291 | . . . . . . 7 |
11 | 10 | neneqd 2226 | . . . . . 6 |
12 | ltpnf 8702 | . . . . . . . . 9 | |
13 | 12 | adantr 261 | . . . . . . . 8 |
14 | breq2 3768 | . . . . . . . . 9 | |
15 | 14 | adantl 262 | . . . . . . . 8 |
16 | 13, 15 | mpbird 156 | . . . . . . 7 |
17 | notnot 559 | . . . . . . . . 9 | |
18 | 17 | olcs 655 | . . . . . . . 8 |
19 | ioran 669 | . . . . . . . 8 | |
20 | 18, 19 | sylnib 601 | . . . . . . 7 |
21 | 16, 20 | syl 14 | . . . . . 6 |
22 | 11, 21 | 2falsed 618 | . . . . 5 |
23 | renemnf 7074 | . . . . . . . . . 10 | |
24 | 23 | adantr 261 | . . . . . . . . 9 |
25 | neeq2 2219 | . . . . . . . . . 10 | |
26 | 25 | adantl 262 | . . . . . . . . 9 |
27 | 24, 26 | mpbird 156 | . . . . . . . 8 |
28 | 27 | necomd 2291 | . . . . . . 7 |
29 | 28 | neneqd 2226 | . . . . . 6 |
30 | mnflt 8704 | . . . . . . . . 9 | |
31 | 30 | adantr 261 | . . . . . . . 8 |
32 | breq1 3767 | . . . . . . . . 9 | |
33 | 32 | adantl 262 | . . . . . . . 8 |
34 | 31, 33 | mpbird 156 | . . . . . . 7 |
35 | orc 633 | . . . . . . 7 | |
36 | oranim 807 | . . . . . . 7 | |
37 | 34, 35, 36 | 3syl 17 | . . . . . 6 |
38 | 29, 37 | 2falsed 618 | . . . . 5 |
39 | 4, 22, 38 | 3jaodan 1201 | . . . 4 |
40 | 39 | ancoms 255 | . . 3 |
41 | renepnf 7073 | . . . . . . . . 9 | |
42 | 41 | adantl 262 | . . . . . . . 8 |
43 | neeq2 2219 | . . . . . . . . 9 | |
44 | 43 | adantr 261 | . . . . . . . 8 |
45 | 42, 44 | mpbird 156 | . . . . . . 7 |
46 | 45 | neneqd 2226 | . . . . . 6 |
47 | ltpnf 8702 | . . . . . . . . 9 | |
48 | 47 | adantl 262 | . . . . . . . 8 |
49 | breq2 3768 | . . . . . . . . 9 | |
50 | 49 | adantr 261 | . . . . . . . 8 |
51 | 48, 50 | mpbird 156 | . . . . . . 7 |
52 | 51, 35, 36 | 3syl 17 | . . . . . 6 |
53 | 46, 52 | 2falsed 618 | . . . . 5 |
54 | eqtr3 2059 | . . . . . . 7 | |
55 | 54 | eqcomd 2045 | . . . . . 6 |
56 | pnfxr 8692 | . . . . . . . . 9 | |
57 | xrltnr 8701 | . . . . . . . . 9 | |
58 | 56, 57 | ax-mp 7 | . . . . . . . 8 |
59 | breq12 3769 | . . . . . . . . 9 | |
60 | 59 | ancoms 255 | . . . . . . . 8 |
61 | 58, 60 | mtbiri 600 | . . . . . . 7 |
62 | breq12 3769 | . . . . . . . 8 | |
63 | 58, 62 | mtbiri 600 | . . . . . . 7 |
64 | 61, 63 | jca 290 | . . . . . 6 |
65 | 55, 64 | 2thd 164 | . . . . 5 |
66 | mnfnepnf 8698 | . . . . . . . . 9 | |
67 | eqeq12 2052 | . . . . . . . . . 10 | |
68 | 67 | necon3bid 2246 | . . . . . . . . 9 |
69 | 66, 68 | mpbiri 157 | . . . . . . . 8 |
70 | 69 | ancoms 255 | . . . . . . 7 |
71 | 70 | neneqd 2226 | . . . . . 6 |
72 | mnfltpnf 8706 | . . . . . . . . 9 | |
73 | breq12 3769 | . . . . . . . . 9 | |
74 | 72, 73 | mpbiri 157 | . . . . . . . 8 |
75 | 74 | ancoms 255 | . . . . . . 7 |
76 | 75, 35, 36 | 3syl 17 | . . . . . 6 |
77 | 71, 76 | 2falsed 618 | . . . . 5 |
78 | 53, 65, 77 | 3jaodan 1201 | . . . 4 |
79 | 78 | ancoms 255 | . . 3 |
80 | renemnf 7074 | . . . . . . . . 9 | |
81 | 80 | adantl 262 | . . . . . . . 8 |
82 | neeq2 2219 | . . . . . . . . 9 | |
83 | 82 | adantr 261 | . . . . . . . 8 |
84 | 81, 83 | mpbird 156 | . . . . . . 7 |
85 | 84 | neneqd 2226 | . . . . . 6 |
86 | mnflt 8704 | . . . . . . . . 9 | |
87 | 86 | adantl 262 | . . . . . . . 8 |
88 | breq1 3767 | . . . . . . . . 9 | |
89 | 88 | adantr 261 | . . . . . . . 8 |
90 | 87, 89 | mpbird 156 | . . . . . . 7 |
91 | 90, 20 | syl 14 | . . . . . 6 |
92 | 85, 91 | 2falsed 618 | . . . . 5 |
93 | 66 | neii 2208 | . . . . . . . . . 10 |
94 | eqeq12 2052 | . . . . . . . . . 10 | |
95 | 93, 94 | mtbiri 600 | . . . . . . . . 9 |
96 | 95 | neneqad 2284 | . . . . . . . 8 |
97 | 96 | necomd 2291 | . . . . . . 7 |
98 | 97 | neneqd 2226 | . . . . . 6 |
99 | breq12 3769 | . . . . . . . 8 | |
100 | 72, 99 | mpbiri 157 | . . . . . . 7 |
101 | 100, 20 | syl 14 | . . . . . 6 |
102 | 98, 101 | 2falsed 618 | . . . . 5 |
103 | eqtr3 2059 | . . . . . . 7 | |
104 | 103 | ancoms 255 | . . . . . 6 |
105 | mnfxr 8694 | . . . . . . . . 9 | |
106 | xrltnr 8701 | . . . . . . . . 9 | |
107 | 105, 106 | ax-mp 7 | . . . . . . . 8 |
108 | breq12 3769 | . . . . . . . . 9 | |
109 | 108 | ancoms 255 | . . . . . . . 8 |
110 | 107, 109 | mtbiri 600 | . . . . . . 7 |
111 | breq12 3769 | . . . . . . . 8 | |
112 | 107, 111 | mtbiri 600 | . . . . . . 7 |
113 | 110, 112 | jca 290 | . . . . . 6 |
114 | 104, 113 | 2thd 164 | . . . . 5 |
115 | 92, 102, 114 | 3jaodan 1201 | . . . 4 |
116 | 115 | ancoms 255 | . . 3 |
117 | 40, 79, 116 | 3jaodan 1201 | . 2 |
118 | 1, 2, 117 | syl2anb 275 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wo 629 w3o 884 wceq 1243 wcel 1393 wne 2204 class class class wbr 3764 cr 6888 cpnf 7057 cmnf 7058 cxr 7059 clt 7060 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-pre-ltirr 6996 ax-pre-apti 6999 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 |
This theorem is referenced by: xrletri3 8721 iccid 8794 |
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