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 7653 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 15295 |
. 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 15296 |
. . . . . . . 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 15296 |
. . . . . . . 8
#
|
| 44 | simpr 110 |
. . . . . . . . 9
# # | |
| 45 | reaplt 8735 |
. . . . . . . . . 10
#
| |
| 46 | 19, 24, 45 | syl2anc 411 |
. . . . . . . . 9
# #
|
| 47 | 44, 46 | mpbid 147 |
. . . . . . . 8
# |
| 48 | 29, 43, 47 | mpjaodan 803 |
. . . . . . 7
# |
| 49 | 48 | inegd 1414 |
. . . . . 6
# |
| 50 | simplrl 535 |
. . . . . . . 8
| |
| 51 | 50 | recnd 8175 |
. . . . . . 7
 |
| 52 | simplrr 536 |
. . . . . . . 8
| |
| 53 | 52 | recnd 8175 |
. . . . . . 7
|
| 54 | apti 8769 |
. . . . . . 7
# | |
| 55 | 51, 53, 54 | syl2anc 411 |
. . . . . 6
# |
| 56 | 49, 55 | mpbird 167 |
. . . . 5
|
| 57 | 56 | ex 115 |
. . . 4
|
| 58 | 57 | ralrimivva 2612 |
. . 3
|
| 59 | breq2 4087 |
. . . . . 6
| |
| 60 | 59 | ralbidv 2530 |
. . . . 5
|
| 61 | breq1 4086 |
. . . . . 6
| |
| 62 | 61 | ralbidv 2530 |
. . . . 5
|
| 63 | 60, 62 | anbi12d 473 |
. . . 4
|
| 64 | 63 | rmo4 2996 |
. . 3
|
| 65 | 58, 64 | sylibr 134 |
. 2
|
| 66 | reu5 2749 |
. 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 713 wceq 1395 wfal 1400 wcel 2200 wral 2508 wrex 2509 wreu 2510 wrmo 2511 cin 3196 wss 3197 c0 3491 class class class wbr 4083
cc 7997
cr 7998
clt 8181
# cap 8728 |
| 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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-cnex 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-mulrcl 8098 ax-addcom 8099 ax-mulcom 8100 ax-addass 8101 ax-mulass 8102 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-1rid 8106 ax-0id 8107 ax-rnegex 8108 ax-precex 8109 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-ltwlin 8112 ax-pre-lttrn 8113 ax-pre-apti 8114 ax-pre-ltadd 8115 ax-pre-mulgt0 8116 ax-pre-suploc 8120 |
| This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-br 4084 df-opab 4146 df-id 4384 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-iota 5278 df-fun 5320 df-fv 5326 df-riota 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-pnf 8183 df-mnf 8184 df-xr 8185 df-ltxr 8186 df-le 8187 df-sub 8319 df-neg 8320 df-reap 8722 df-ap 8729 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |