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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 7274 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 12768 |
. 2
|
| 10 | 1 | ad4antr 485 |
. . . . . . . . 9
#
|
| 11 | 2 | ad4antr 485 |
. . . . . . . . 9
#
|
| 12 | 3 | ad4antr 485 |
. . . . . . . . 9
#
|
| 13 | 4 | ad4antr 485 |
. . . . . . . . 9
#
|
| 14 | 5 | ad4antr 485 |
. . . . . . . . 9
#
|
| 15 | 6 | ad4antr 485 |
. . . . . . . . 9
#
|
| 16 | 7 | ad4antr 485 |
. . . . . . . . 9
#
|
| 17 | 8 | ad4antr 485 |
. . . . . . . . 9
#
|
| 18 | simprl 520 |
. . . . . . . . . . 11
| |
| 19 | 18 | ad2antrr 479 |
. . . . . . . . . 10
# |
| 20 | 19 | adantr 274 |
. . . . . . . . 9
#
|
| 21 | simprl 520 |
. . . . . . . . . 10
| |
| 22 | 21 | ad2antrr 479 |
. . . . . . . . 9
#
|
| 23 | simprr 521 |
. . . . . . . . . . 11
| |
| 24 | 23 | ad2antrr 479 |
. . . . . . . . . 10
# |
| 25 | 24 | adantr 274 |
. . . . . . . . 9
#
|
| 26 | simprr 521 |
. . . . . . . . . 10
| |
| 27 | 26 | ad2antrr 479 |
. . . . . . . . 9
#
|
| 28 | simpr 109 |
. . . . . . . . 9
#
| |
| 29 | 10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 25, 27, 28 | dedekindeulemeu 12769 |
. . . . . . . 8
#
 |
| 30 | 1 | ad4antr 485 |
. . . . . . . . 9
#
|
| 31 | 2 | ad4antr 485 |
. . . . . . . . 9
#
|
| 32 | 3 | ad4antr 485 |
. . . . . . . . 9
#
|
| 33 | 4 | ad4antr 485 |
. . . . . . . . 9
#
|
| 34 | 5 | ad4antr 485 |
. . . . . . . . 9
#
|
| 35 | 6 | ad4antr 485 |
. . . . . . . . 9
#
|
| 36 | 7 | ad4antr 485 |
. . . . . . . . 9
#
|
| 37 | 8 | ad4antr 485 |
. . . . . . . . 9
#
|
| 38 | 24 | adantr 274 |
. . . . . . . . 9
#
|
| 39 | 26 | ad2antrr 479 |
. . . . . . . . 9
#
|
| 40 | 19 | adantr 274 |
. . . . . . . . 9
#
|
| 41 | 21 | ad2antrr 479 |
. . . . . . . . 9
#
|
| 42 | simpr 109 |
. . . . . . . . 9
#
| |
| 43 | 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42 | dedekindeulemeu 12769 |
. . . . . . . 8
#
|
| 44 | simpr 109 |
. . . . . . . . 9
# # | |
| 45 | reaplt 8350 |
. . . . . . . . . 10
#
| |
| 46 | 19, 24, 45 | syl2anc 408 |
. . . . . . . . 9
# #
|
| 47 | 44, 46 | mpbid 146 |
. . . . . . . 8
# |
| 48 | 29, 43, 47 | mpjaodan 787 |
. . . . . . 7
# |
| 49 | 48 | inegd 1350 |
. . . . . 6
# |
| 50 | simplrl 524 |
. . . . . . . 8
| |
| 51 | 50 | recnd 7794 |
. . . . . . 7
 |
| 52 | simplrr 525 |
. . . . . . . 8
| |
| 53 | 52 | recnd 7794 |
. . . . . . 7
|
| 54 | apti 8384 |
. . . . . . 7
# | |
| 55 | 51, 53, 54 | syl2anc 408 |
. . . . . 6
# |
| 56 | 49, 55 | mpbird 166 |
. . . . 5
|
| 57 | 56 | ex 114 |
. . . 4
|
| 58 | 57 | ralrimivva 2514 |
. . 3
|
| 59 | breq2 3933 |
. . . . . 6
| |
| 60 | 59 | ralbidv 2437 |
. . . . 5
|
| 61 | breq1 3932 |
. . . . . 6
| |
| 62 | 61 | ralbidv 2437 |
. . . . 5
|
| 63 | 60, 62 | anbi12d 464 |
. . . 4
|
| 64 | 63 | rmo4 2877 |
. . 3
|
| 65 | 58, 64 | sylibr 133 |
. 2
|
| 66 | reu5 2643 |
. 2
| |
| 67 | 9, 65, 66 | sylanbrc 413 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wn 3
wi 4
wa 103
wb 104 wo 697 wceq 1331 wfal 1336 wcel 1480 wral 2416 wrex 2417 wreu 2418 wrmo 2419 cin 3070 wss 3071 c0 3363 class class class wbr 3929
cc 7618
cr 7619
clt 7800
# cap 8343 |
| 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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-suploc 7741 |
| This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 |
| This theorem is referenced by: (None) |
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