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 7428 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 13393 |
. 2
|
| 10 | 1 | ad4antr 491 |
. . . . . . . . 9
#
|
| 11 | 2 | ad4antr 491 |
. . . . . . . . 9
#
|
| 12 | 3 | ad4antr 491 |
. . . . . . . . 9
#
|
| 13 | 4 | ad4antr 491 |
. . . . . . . . 9
#
|
| 14 | 5 | ad4antr 491 |
. . . . . . . . 9
#
|
| 15 | 6 | ad4antr 491 |
. . . . . . . . 9
#
|
| 16 | 7 | ad4antr 491 |
. . . . . . . . 9
#
|
| 17 | 8 | ad4antr 491 |
. . . . . . . . 9
#
|
| 18 | simprl 526 |
. . . . . . . . . . 11
| |
| 19 | 18 | ad2antrr 485 |
. . . . . . . . . 10
# |
| 20 | 19 | adantr 274 |
. . . . . . . . 9
#
|
| 21 | simprl 526 |
. . . . . . . . . 10
| |
| 22 | 21 | ad2antrr 485 |
. . . . . . . . 9
#
|
| 23 | simprr 527 |
. . . . . . . . . . 11
| |
| 24 | 23 | ad2antrr 485 |
. . . . . . . . . 10
# |
| 25 | 24 | adantr 274 |
. . . . . . . . 9
#
|
| 26 | simprr 527 |
. . . . . . . . . 10
| |
| 27 | 26 | ad2antrr 485 |
. . . . . . . . 9
#
|
| 28 | simpr 109 |
. . . . . . . . 9
#
| |
| 29 | 10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 25, 27, 28 | dedekindeulemeu 13394 |
. . . . . . . 8
#
 |
| 30 | 1 | ad4antr 491 |
. . . . . . . . 9
#
|
| 31 | 2 | ad4antr 491 |
. . . . . . . . 9
#
|
| 32 | 3 | ad4antr 491 |
. . . . . . . . 9
#
|
| 33 | 4 | ad4antr 491 |
. . . . . . . . 9
#
|
| 34 | 5 | ad4antr 491 |
. . . . . . . . 9
#
|
| 35 | 6 | ad4antr 491 |
. . . . . . . . 9
#
|
| 36 | 7 | ad4antr 491 |
. . . . . . . . 9
#
|
| 37 | 8 | ad4antr 491 |
. . . . . . . . 9
#
|
| 38 | 24 | adantr 274 |
. . . . . . . . 9
#
|
| 39 | 26 | ad2antrr 485 |
. . . . . . . . 9
#
|
| 40 | 19 | adantr 274 |
. . . . . . . . 9
#
|
| 41 | 21 | ad2antrr 485 |
. . . . . . . . 9
#
|
| 42 | simpr 109 |
. . . . . . . . 9
#
| |
| 43 | 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42 | dedekindeulemeu 13394 |
. . . . . . . 8
#
|
| 44 | simpr 109 |
. . . . . . . . 9
# # | |
| 45 | reaplt 8507 |
. . . . . . . . . 10
#
| |
| 46 | 19, 24, 45 | syl2anc 409 |
. . . . . . . . 9
# #
|
| 47 | 44, 46 | mpbid 146 |
. . . . . . . 8
# |
| 48 | 29, 43, 47 | mpjaodan 793 |
. . . . . . 7
# |
| 49 | 48 | inegd 1367 |
. . . . . 6
# |
| 50 | simplrl 530 |
. . . . . . . 8
| |
| 51 | 50 | recnd 7948 |
. . . . . . 7
 |
| 52 | simplrr 531 |
. . . . . . . 8
| |
| 53 | 52 | recnd 7948 |
. . . . . . 7
|
| 54 | apti 8541 |
. . . . . . 7
# | |
| 55 | 51, 53, 54 | syl2anc 409 |
. . . . . 6
# |
| 56 | 49, 55 | mpbird 166 |
. . . . 5
|
| 57 | 56 | ex 114 |
. . . 4
|
| 58 | 57 | ralrimivva 2552 |
. . 3
|
| 59 | breq2 3993 |
. . . . . 6
| |
| 60 | 59 | ralbidv 2470 |
. . . . 5
|
| 61 | breq1 3992 |
. . . . . 6
| |
| 62 | 61 | ralbidv 2470 |
. . . . 5
|
| 63 | 60, 62 | anbi12d 470 |
. . . 4
|
| 64 | 63 | rmo4 2923 |
. . 3
|
| 65 | 58, 64 | sylibr 133 |
. 2
|
| 66 | reu5 2682 |
. 2
| |
| 67 | 9, 65, 66 | sylanbrc 415 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wn 3
wi 4
wa 103
wb 104 wo 703 wceq 1348 wfal 1353 wcel 2141 wral 2448 wrex 2449 wreu 2450 wrmo 2451 cin 3120 wss 3121 c0 3414 class class class wbr 3989
cc 7772
cr 7773
clt 7954
# cap 8500 |
| 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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-suploc 7895 |
| This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |