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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 7456 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 13766 |
. 2
|
| 10 | 1 | ad4antr 494 |
. . . . . . . . 9
#
|
| 11 | 2 | ad4antr 494 |
. . . . . . . . 9
#
|
| 12 | 3 | ad4antr 494 |
. . . . . . . . 9
#
|
| 13 | 4 | ad4antr 494 |
. . . . . . . . 9
#
|
| 14 | 5 | ad4antr 494 |
. . . . . . . . 9
#
|
| 15 | 6 | ad4antr 494 |
. . . . . . . . 9
#
|
| 16 | 7 | ad4antr 494 |
. . . . . . . . 9
#
|
| 17 | 8 | ad4antr 494 |
. . . . . . . . 9
#
|
| 18 | simprl 529 |
. . . . . . . . . . 11
| |
| 19 | 18 | ad2antrr 488 |
. . . . . . . . . 10
# |
| 20 | 19 | adantr 276 |
. . . . . . . . 9
#
|
| 21 | simprl 529 |
. . . . . . . . . 10
| |
| 22 | 21 | ad2antrr 488 |
. . . . . . . . 9
#
|
| 23 | simprr 531 |
. . . . . . . . . . 11
| |
| 24 | 23 | ad2antrr 488 |
. . . . . . . . . 10
# |
| 25 | 24 | adantr 276 |
. . . . . . . . 9
#
|
| 26 | simprr 531 |
. . . . . . . . . 10
| |
| 27 | 26 | ad2antrr 488 |
. . . . . . . . 9
#
|
| 28 | simpr 110 |
. . . . . . . . 9
#
| |
| 29 | 10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 25, 27, 28 | dedekindeulemeu 13767 |
. . . . . . . 8
#
 |
| 30 | 1 | ad4antr 494 |
. . . . . . . . 9
#
|
| 31 | 2 | ad4antr 494 |
. . . . . . . . 9
#
|
| 32 | 3 | ad4antr 494 |
. . . . . . . . 9
#
|
| 33 | 4 | ad4antr 494 |
. . . . . . . . 9
#
|
| 34 | 5 | ad4antr 494 |
. . . . . . . . 9
#
|
| 35 | 6 | ad4antr 494 |
. . . . . . . . 9
#
|
| 36 | 7 | ad4antr 494 |
. . . . . . . . 9
#
|
| 37 | 8 | ad4antr 494 |
. . . . . . . . 9
#
|
| 38 | 24 | adantr 276 |
. . . . . . . . 9
#
|
| 39 | 26 | ad2antrr 488 |
. . . . . . . . 9
#
|
| 40 | 19 | adantr 276 |
. . . . . . . . 9
#
|
| 41 | 21 | ad2antrr 488 |
. . . . . . . . 9
#
|
| 42 | simpr 110 |
. . . . . . . . 9
#
| |
| 43 | 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42 | dedekindeulemeu 13767 |
. . . . . . . 8
#
|
| 44 | simpr 110 |
. . . . . . . . 9
# # | |
| 45 | reaplt 8535 |
. . . . . . . . . 10
#
| |
| 46 | 19, 24, 45 | syl2anc 411 |
. . . . . . . . 9
# #
|
| 47 | 44, 46 | mpbid 147 |
. . . . . . . 8
# |
| 48 | 29, 43, 47 | mpjaodan 798 |
. . . . . . 7
# |
| 49 | 48 | inegd 1372 |
. . . . . 6
# |
| 50 | simplrl 535 |
. . . . . . . 8
| |
| 51 | 50 | recnd 7976 |
. . . . . . 7
 |
| 52 | simplrr 536 |
. . . . . . . 8
| |
| 53 | 52 | recnd 7976 |
. . . . . . 7
|
| 54 | apti 8569 |
. . . . . . 7
# | |
| 55 | 51, 53, 54 | syl2anc 411 |
. . . . . 6
# |
| 56 | 49, 55 | mpbird 167 |
. . . . 5
|
| 57 | 56 | ex 115 |
. . . 4
|
| 58 | 57 | ralrimivva 2559 |
. . 3
|
| 59 | breq2 4004 |
. . . . . 6
| |
| 60 | 59 | ralbidv 2477 |
. . . . 5
|
| 61 | breq1 4003 |
. . . . . 6
| |
| 62 | 61 | ralbidv 2477 |
. . . . 5
|
| 63 | 60, 62 | anbi12d 473 |
. . . 4
|
| 64 | 63 | rmo4 2930 |
. . 3
|
| 65 | 58, 64 | sylibr 134 |
. 2
|
| 66 | reu5 2689 |
. 2
| |
| 67 | 9, 65, 66 | sylanbrc 417 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wn 3
wi 4
wa 104
wb 105 wo 708 wceq 1353 wfal 1358 wcel 2148 wral 2455 wrex 2456 wreu 2457 wrmo 2458 cin 3128 wss 3129 c0 3422 class class class wbr 4000
cc 7800
cr 7801
clt 7982
# cap 8528 |
| 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 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-suploc 7923 |
| This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-iota 5174 df-fun 5214 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 |
| This theorem is referenced by: (None) |
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