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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 7833 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 15722 |
. 2
|
| 10 | 1 | ad4antr 498 |
. . . . . . . . 9
#
|
| 11 | 2 | ad4antr 498 |
. . . . . . . . 9
#
|
| 12 | 3 | ad4antr 498 |
. . . . . . . . 9
#
|
| 13 | 4 | ad4antr 498 |
. . . . . . . . 9
#
|
| 14 | 5 | ad4antr 498 |
. . . . . . . . 9
#
|
| 15 | 6 | ad4antr 498 |
. . . . . . . . 9
#
|
| 16 | 7 | ad4antr 498 |
. . . . . . . . 9
#
|
| 17 | 8 | ad4antr 498 |
. . . . . . . . 9
#
|
| 18 | simprl 535 |
. . . . . . . . . . 11
| |
| 19 | 18 | ad2antrr 492 |
. . . . . . . . . 10
# |
| 20 | 19 | adantr 276 |
. . . . . . . . 9
#
|
| 21 | simprl 535 |
. . . . . . . . . 10
| |
| 22 | 21 | ad2antrr 492 |
. . . . . . . . 9
#
|
| 23 | simprr 537 |
. . . . . . . . . . 11
| |
| 24 | 23 | ad2antrr 492 |
. . . . . . . . . 10
# |
| 25 | 24 | adantr 276 |
. . . . . . . . 9
#
|
| 26 | simprr 537 |
. . . . . . . . . 10
| |
| 27 | 26 | ad2antrr 492 |
. . . . . . . . 9
#
|
| 28 | simpr 110 |
. . . . . . . . 9
#
| |
| 29 | 10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 25, 27, 28 | dedekindeulemeu 15723 |
. . . . . . . 8
#
 |
| 30 | 1 | ad4antr 498 |
. . . . . . . . 9
#
|
| 31 | 2 | ad4antr 498 |
. . . . . . . . 9
#
|
| 32 | 3 | ad4antr 498 |
. . . . . . . . 9
#
|
| 33 | 4 | ad4antr 498 |
. . . . . . . . 9
#
|
| 34 | 5 | ad4antr 498 |
. . . . . . . . 9
#
|
| 35 | 6 | ad4antr 498 |
. . . . . . . . 9
#
|
| 36 | 7 | ad4antr 498 |
. . . . . . . . 9
#
|
| 37 | 8 | ad4antr 498 |
. . . . . . . . 9
#
|
| 38 | 24 | adantr 276 |
. . . . . . . . 9
#
|
| 39 | 26 | ad2antrr 492 |
. . . . . . . . 9
#
|
| 40 | 19 | adantr 276 |
. . . . . . . . 9
#
|
| 41 | 21 | ad2antrr 492 |
. . . . . . . . 9
#
|
| 42 | simpr 110 |
. . . . . . . . 9
#
| |
| 43 | 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42 | dedekindeulemeu 15723 |
. . . . . . . 8
#
|
| 44 | simpr 110 |
. . . . . . . . 9
# # | |
| 45 | reaplt 8916 |
. . . . . . . . . 10
#
| |
| 46 | 19, 24, 45 | syl2anc 415 |
. . . . . . . . 9
# #
|
| 47 | 44, 46 | mpbid 147 |
. . . . . . . 8
# |
| 48 | 29, 43, 47 | mpjaodan 810 |
. . . . . . 7
# |
| 49 | 48 | inegd 1421 |
. . . . . 6
# |
| 50 | simplrl 541 |
. . . . . . . 8
| |
| 51 | 50 | recnd 8354 |
. . . . . . 7
 |
| 52 | simplrr 542 |
. . . . . . . 8
| |
| 53 | 52 | recnd 8354 |
. . . . . . 7
|
| 54 | apti 8950 |
. . . . . . 7
# | |
| 55 | 51, 53, 54 | syl2anc 415 |
. . . . . 6
# |
| 56 | 49, 55 | mpbird 167 |
. . . . 5
|
| 57 | 56 | ex 115 |
. . . 4
|
| 58 | 57 | ralrimivva 2632 |
. . 3
|
| 59 | breq2 4134 |
. . . . . 6
| |
| 60 | 59 | ralbidv 2550 |
. . . . 5
|
| 61 | breq1 4133 |
. . . . . 6
| |
| 62 | 61 | ralbidv 2550 |
. . . . 5
|
| 63 | 60, 62 | anbi12d 477 |
. . . 4
|
| 64 | 63 | rmo4 3019 |
. . 3
|
| 65 | 58, 64 | sylibr 134 |
. 2
|
| 66 | reu5 2770 |
. 2
| |
| 67 | 9, 65, 66 | sylanbrc 421 |
1
|
| Colors of variables: wff set class |
| This proof depends on syntax axioms:
wn 3
wi 4
wa 104
wb 105 wo 720 wceq 1402 wfal 1407 wcel 2209 wral 2528 wrex 2529 wreu 2530 wrmo 2531 cin 3219 wss 3220 c0 3520 class class class wbr 4130
cc 8177
cr 8178
clt 8360
# cap 8909 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-suploc 8300 |
| This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-br 4131 df-opab 4193 df-id 4438 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-iota 5337 df-fun 5379 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910 |
| This theorem is used by: (None) |
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