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| Mirrors > Home > ILE Home > Th. List > dedekindeu | Unicode version | ||
| Description: A Dedekind cut identifies a unique real number. Similar to df-inp 7407 except that the 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, 5-Jan-2024.) |
| Ref | Expression |
|---|---|
| dedekindeu.lss |
     
  |
| dedekindeu.uss |
|
| dedekindeu.lm |
  
|
| dedekindeu.um |
|
| dedekindeu.lr |
   |
| dedekindeu.ur |
|
| dedekindeu.disj |
   |
| dedekindeu.loc |
|
| Ref | Expression |
|---|---|
| dedekindeu |
  ![/\](wedge.gif "/\"){kind=small}
|
,,, ,,, ,,,
is distinct from all other
variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dedekindeu.lss |
. . 3
| |
| 2 | dedekindeu.uss |
. . 3
| |
| 3 | dedekindeu.lm |
. . 3
| |
| 4 | dedekindeu.um |
. . 3
| |
| 5 | dedekindeu.lr |
. . 3
| |
| 6 | dedekindeu.ur |
. . 3
| |
| 7 | dedekindeu.disj |
. . 3
| |
| 8 | dedekindeu.loc |
. . 3
| |
| 9 | 1, 2, 3, 4, 5, 6, 7, 8 | dedekindeulemlu 13239 |
. 2
|
| 10 | 1 | ad4antr 486 |
. . . . . . . . 9
#
|
| 11 | 2 | ad4antr 486 |
. . . . . . . . 9
#
|
| 12 | 3 | ad4antr 486 |
. . . . . . . . 9
#
|
| 13 | 4 | ad4antr 486 |
. . . . . . . . 9
#
|
| 14 | 5 | ad4antr 486 |
. . . . . . . . 9
#
|
| 15 | 6 | ad4antr 486 |
. . . . . . . . 9
#
|
| 16 | 7 | ad4antr 486 |
. . . . . . . . 9
#
|
| 17 | 8 | ad4antr 486 |
. . . . . . . . 9
#
|
| 18 | simprl 521 |
. . . . . . . . . . 11
| |
| 19 | 18 | ad2antrr 480 |
. . . . . . . . . 10
# |
| 20 | 19 | adantr 274 |
. . . . . . . . 9
#
|
| 21 | simprl 521 |
. . . . . . . . . 10
| |
| 22 | 21 | ad2antrr 480 |
. . . . . . . . 9
#
|
| 23 | simprr 522 |
. . . . . . . . . . 11
| |
| 24 | 23 | ad2antrr 480 |
. . . . . . . . . 10
# |
| 25 | 24 | adantr 274 |
. . . . . . . . 9
#
|
| 26 | simprr 522 |
. . . . . . . . . 10
| |
| 27 | 26 | ad2antrr 480 |
. . . . . . . . 9
#
|
| 28 | simpr 109 |
. . . . . . . . 9
#
| |
| 29 | 10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 25, 27, 28 | dedekindeulemeu 13240 |
. . . . . . . 8
#
 |
| 30 | 1 | ad4antr 486 |
. . . . . . . . 9
#
|
| 31 | 2 | ad4antr 486 |
. . . . . . . . 9
#
|
| 32 | 3 | ad4antr 486 |
. . . . . . . . 9
#
|
| 33 | 4 | ad4antr 486 |
. . . . . . . . 9
#
|
| 34 | 5 | ad4antr 486 |
. . . . . . . . 9
#
|
| 35 | 6 | ad4antr 486 |
. . . . . . . . 9
#
|
| 36 | 7 | ad4antr 486 |
. . . . . . . . 9
#
|
| 37 | 8 | ad4antr 486 |
. . . . . . . . 9
#
|
| 38 | 24 | adantr 274 |
. . . . . . . . 9
#
|
| 39 | 26 | ad2antrr 480 |
. . . . . . . . 9
#
|
| 40 | 19 | adantr 274 |
. . . . . . . . 9
#
|
| 41 | 21 | ad2antrr 480 |
. . . . . . . . 9
#
|
| 42 | simpr 109 |
. . . . . . . . 9
#
| |
| 43 | 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42 | dedekindeulemeu 13240 |
. . . . . . . 8
#
|
| 44 | simpr 109 |
. . . . . . . . 9
# # | |
| 45 | reaplt 8486 |
. . . . . . . . . 10
#
| |
| 46 | 19, 24, 45 | syl2anc 409 |
. . . . . . . . 9
# #
|
| 47 | 44, 46 | mpbid 146 |
. . . . . . . 8
# |
| 48 | 29, 43, 47 | mpjaodan 788 |
. . . . . . 7
# |
| 49 | 48 | inegd 1362 |
. . . . . 6
# |
| 50 | simplrl 525 |
. . . . . . . 8
| |
| 51 | 50 | recnd 7927 |
. . . . . . 7
 |
| 52 | simplrr 526 |
. . . . . . . 8
| |
| 53 | 52 | recnd 7927 |
. . . . . . 7
|
| 54 | apti 8520 |
. . . . . . 7
# | |
| 55 | 51, 53, 54 | syl2anc 409 |
. . . . . 6
# |
| 56 | 49, 55 | mpbird 166 |
. . . . 5
|
| 57 | 56 | ex 114 |
. . . 4
|
| 58 | 57 | ralrimivva 2548 |
. . 3
|
| 59 | breq2 3986 |
. . . . . 6
| |
| 60 | 59 | ralbidv 2466 |
. . . . 5
|
| 61 | breq1 3985 |
. . . . . 6
| |
| 62 | 61 | ralbidv 2466 |
. . . . 5
|
| 63 | 60, 62 | anbi12d 465 |
. . . 4
|
| 64 | 63 | rmo4 2919 |
. . 3
|
| 65 | 58, 64 | sylibr 133 |
. 2
|
| 66 | reu5 2678 |
. 2
| |
| 67 | 9, 65, 66 | sylanbrc 414 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wn 3
wi 4
wa 103
wb 104 wo 698 wceq 1343 wfal 1348 wcel 2136 wral 2444 wrex 2445 wreu 2446 wrmo 2447 cin 3115 wss 3116 c0 3409 class class class wbr 3982
cc 7751
cr 7752
clt 7933
# cap 8479 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-suploc 7874 |
| This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-iota 5153 df-fun 5190 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 |
| This theorem is referenced by: (None) |
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