Nearby theorems
|
|
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > ILE Home > Th. List > dedekindeu | Unicode version | ||
| Description: A Dedekind cut identifies a unique real number. Similar to df-inp 7386 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 13010 |
. 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 13011 |
. . . . . . . 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 13011 |
. . . . . . . 8
#
|
| 44 | simpr 109 |
. . . . . . . . 9
# # | |
| 45 | reaplt 8463 |
. . . . . . . . . 10
#
| |
| 46 | 19, 24, 45 | syl2anc 409 |
. . . . . . . . 9
# #
|
| 47 | 44, 46 | mpbid 146 |
. . . . . . . 8
# |
| 48 | 29, 43, 47 | mpjaodan 788 |
. . . . . . 7
# |
| 49 | 48 | inegd 1354 |
. . . . . 6
# |
| 50 | simplrl 525 |
. . . . . . . 8
| |
| 51 | 50 | recnd 7906 |
. . . . . . 7
 |
| 52 | simplrr 526 |
. . . . . . . 8
| |
| 53 | 52 | recnd 7906 |
. . . . . . 7
|
| 54 | apti 8497 |
. . . . . . 7
# | |
| 55 | 51, 53, 54 | syl2anc 409 |
. . . . . 6
# |
| 56 | 49, 55 | mpbird 166 |
. . . . 5
|
| 57 | 56 | ex 114 |
. . . 4
|
| 58 | 57 | ralrimivva 2539 |
. . 3
|
| 59 | breq2 3969 |
. . . . . 6
| |
| 60 | 59 | ralbidv 2457 |
. . . . 5
|
| 61 | breq1 3968 |
. . . . . 6
| |
| 62 | 61 | ralbidv 2457 |
. . . . 5
|
| 63 | 60, 62 | anbi12d 465 |
. . . 4
|
| 64 | 63 | rmo4 2905 |
. . 3
|
| 65 | 58, 64 | sylibr 133 |
. 2
|
| 66 | reu5 2669 |
. 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 1335 wfal 1340 wcel 2128 wral 2435 wrex 2436 wreu 2437 wrmo 2438 cin 3101 wss 3102 c0 3394 class class class wbr 3965
cc 7730
cr 7731
clt 7912
# cap 8456 |
| 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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 ax-pre-suploc 7853 |
| This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-iota 5135 df-fun 5172 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |