Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > dedekindicc | Unicode version |
Description: A Dedekind cut identifies a unique real number. Similar to df-inp 7440 except that the Dedekind cut is formed by sets of reals (rather than positive rationals). But in both cases the defining property of a Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and located. (Contributed by Jim Kingdon, 19-Feb-2024.) |
Ref | Expression |
---|---|
dedekindicc.a | |
dedekindicc.b | |
dedekindicc.lss | |
dedekindicc.uss | |
dedekindicc.lm | |
dedekindicc.um | |
dedekindicc.lr | |
dedekindicc.ur | |
dedekindicc.disj | |
dedekindicc.loc | |
dedekindicc.ab |
Ref | Expression |
---|---|
dedekindicc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dedekindicc.a | . . . . 5 | |
2 | dedekindicc.b | . . . . 5 | |
3 | dedekindicc.lss | . . . . 5 | |
4 | dedekindicc.uss | . . . . 5 | |
5 | dedekindicc.lm | . . . . 5 | |
6 | dedekindicc.um | . . . . 5 | |
7 | dedekindicc.lr | . . . . 5 | |
8 | dedekindicc.ur | . . . . 5 | |
9 | dedekindicc.disj | . . . . 5 | |
10 | dedekindicc.loc | . . . . 5 | |
11 | dedekindicc.ab | . . . . 5 | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | dedekindicclemicc 13690 | . . . 4 |
13 | df-reu 2460 | . . . 4 | |
14 | 12, 13 | sylib 122 | . . 3 |
15 | breq1 4001 | . . . . . . . . . 10 | |
16 | 15 | cbvralv 2701 | . . . . . . . . 9 |
17 | breq2 4002 | . . . . . . . . . 10 | |
18 | 17 | cbvralv 2701 | . . . . . . . . 9 |
19 | 16, 18 | anbi12i 460 | . . . . . . . 8 |
20 | 19 | anbi2i 457 | . . . . . . 7 |
21 | iccssre 9926 | . . . . . . . . . . 11 | |
22 | 1, 2, 21 | syl2anc 411 | . . . . . . . . . 10 |
23 | 22 | sselda 3153 | . . . . . . . . 9 |
24 | 23 | adantrr 479 | . . . . . . . 8 |
25 | 5 | adantr 276 | . . . . . . . . 9 |
26 | 1 | ad2antrr 488 | . . . . . . . . . 10 |
27 | simpll 527 | . . . . . . . . . . 11 | |
28 | simprl 529 | . . . . . . . . . . 11 | |
29 | 22 | sseld 3152 | . . . . . . . . . . 11 |
30 | 27, 28, 29 | sylc 62 | . . . . . . . . . 10 |
31 | 24 | adantr 276 | . . . . . . . . . 10 |
32 | 1 | rexrd 7981 | . . . . . . . . . . . 12 |
33 | 32 | ad2antrr 488 | . . . . . . . . . . 11 |
34 | 2 | rexrd 7981 | . . . . . . . . . . . 12 |
35 | 34 | ad2antrr 488 | . . . . . . . . . . 11 |
36 | iccgelb 9903 | . . . . . . . . . . 11 | |
37 | 33, 35, 28, 36 | syl3anc 1238 | . . . . . . . . . 10 |
38 | breq1 4001 | . . . . . . . . . . 11 | |
39 | simprrl 539 | . . . . . . . . . . . 12 | |
40 | 39 | adantr 276 | . . . . . . . . . . 11 |
41 | simprr 531 | . . . . . . . . . . 11 | |
42 | 38, 40, 41 | rspcdva 2844 | . . . . . . . . . 10 |
43 | 26, 30, 31, 37, 42 | lelttrd 8056 | . . . . . . . . 9 |
44 | 25, 43 | rexlimddv 2597 | . . . . . . . 8 |
45 | 6 | adantr 276 | . . . . . . . . 9 |
46 | 24 | adantr 276 | . . . . . . . . . 10 |
47 | simpll 527 | . . . . . . . . . . 11 | |
48 | simprl 529 | . . . . . . . . . . 11 | |
49 | 22 | sseld 3152 | . . . . . . . . . . 11 |
50 | 47, 48, 49 | sylc 62 | . . . . . . . . . 10 |
51 | 2 | ad2antrr 488 | . . . . . . . . . 10 |
52 | breq2 4002 | . . . . . . . . . . 11 | |
53 | simprrr 540 | . . . . . . . . . . . 12 | |
54 | 53 | adantr 276 | . . . . . . . . . . 11 |
55 | simprr 531 | . . . . . . . . . . 11 | |
56 | 52, 54, 55 | rspcdva 2844 | . . . . . . . . . 10 |
57 | 32 | ad2antrr 488 | . . . . . . . . . . 11 |
58 | 34 | ad2antrr 488 | . . . . . . . . . . 11 |
59 | iccleub 9902 | . . . . . . . . . . 11 | |
60 | 57, 58, 48, 59 | syl3anc 1238 | . . . . . . . . . 10 |
61 | 46, 50, 51, 56, 60 | ltletrd 8354 | . . . . . . . . 9 |
62 | 45, 61 | rexlimddv 2597 | . . . . . . . 8 |
63 | 32 | adantr 276 | . . . . . . . . 9 |
64 | 34 | adantr 276 | . . . . . . . . 9 |
65 | elioo2 9892 | . . . . . . . . 9 | |
66 | 63, 64, 65 | syl2anc 411 | . . . . . . . 8 |
67 | 24, 44, 62, 66 | mpbir3and 1180 | . . . . . . 7 |
68 | 20, 67 | sylan2b 287 | . . . . . 6 |
69 | simprr 531 | . . . . . 6 | |
70 | 68, 69 | jca 306 | . . . . 5 |
71 | ioossicc 9930 | . . . . . . . 8 | |
72 | 71 | sseli 3149 | . . . . . . 7 |
73 | 72 | ad2antrl 490 | . . . . . 6 |
74 | simprr 531 | . . . . . 6 | |
75 | 73, 74 | jca 306 | . . . . 5 |
76 | 70, 75 | impbida 596 | . . . 4 |
77 | 76 | eubidv 2032 | . . 3 |
78 | 14, 77 | mpbid 147 | . 2 |
79 | df-reu 2460 | . 2 | |
80 | 78, 79 | sylibr 134 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 wo 708 w3a 978 wceq 1353 weu 2024 wcel 2146 wral 2453 wrex 2454 wreu 2455 cin 3126 wss 3127 c0 3420 class class class wbr 3998 (class class class)co 5865 cr 7785 cxr 7965 clt 7966 cle 7967 cioo 9859 cicc 9862 |
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 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 ax-pre-mulext 7904 ax-arch 7905 ax-caucvg 7906 ax-pre-suploc 7907 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rmo 2461 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-po 4290 df-iso 4291 df-iord 4360 df-on 4362 df-ilim 4363 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-isom 5217 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-frec 6382 df-sup 6973 df-inf 6974 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-reap 8506 df-ap 8513 df-div 8603 df-inn 8893 df-2 8951 df-3 8952 df-4 8953 df-n0 9150 df-z 9227 df-uz 9502 df-rp 9625 df-ioo 9863 df-icc 9866 df-seqfrec 10416 df-exp 10490 df-cj 10819 df-re 10820 df-im 10821 df-rsqrt 10975 df-abs 10976 |
This theorem is referenced by: ivthinclemex 13700 |
Copyright terms: Public domain | W3C validator |