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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 7823 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 15645 |
. 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 15646 |
. . . . . . . 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 15646 |
. . . . . . . 8
#
|
| 44 | simpr 110 |
. . . . . . . . 9
# # | |
| 45 | reaplt 8906 |
. . . . . . . . . 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 8344 |
. . . . . . 7
 |
| 52 | simplrr 542 |
. . . . . . . 8
| |
| 53 | 52 | recnd 8344 |
. . . . . . 7
|
| 54 | apti 8940 |
. . . . . . 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 4129 |
. . . . . 6
| |
| 60 | 59 | ralbidv 2550 |
. . . . 5
|
| 61 | breq1 4128 |
. . . . . 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 |
| Syntax hints: 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 4125
cc 8167
cr 8168
clt 8350
# cap 8899 |
| 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 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 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-suploc 8290 |
| This theorem 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 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-br 4126 df-opab 4188 df-id 4433 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-iota 5332 df-fun 5374 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 |
| This theorem is referenced by: (None) |
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