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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 7550 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 14941 |
. 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 14942 |
. . . . . . . 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 14942 |
. . . . . . . 8
#
|
| 44 | simpr 110 |
. . . . . . . . 9
# # | |
| 45 | reaplt 8632 |
. . . . . . . . . 10
#
| |
| 46 | 19, 24, 45 | syl2anc 411 |
. . . . . . . . 9
# #
|
| 47 | 44, 46 | mpbid 147 |
. . . . . . . 8
# |
| 48 | 29, 43, 47 | mpjaodan 799 |
. . . . . . 7
# |
| 49 | 48 | inegd 1383 |
. . . . . 6
# |
| 50 | simplrl 535 |
. . . . . . . 8
| |
| 51 | 50 | recnd 8072 |
. . . . . . 7
 |
| 52 | simplrr 536 |
. . . . . . . 8
| |
| 53 | 52 | recnd 8072 |
. . . . . . 7
|
| 54 | apti 8666 |
. . . . . . 7
# | |
| 55 | 51, 53, 54 | syl2anc 411 |
. . . . . 6
# |
| 56 | 49, 55 | mpbird 167 |
. . . . 5
|
| 57 | 56 | ex 115 |
. . . 4
|
| 58 | 57 | ralrimivva 2579 |
. . 3
|
| 59 | breq2 4038 |
. . . . . 6
| |
| 60 | 59 | ralbidv 2497 |
. . . . 5
|
| 61 | breq1 4037 |
. . . . . 6
| |
| 62 | 61 | ralbidv 2497 |
. . . . 5
|
| 63 | 60, 62 | anbi12d 473 |
. . . 4
|
| 64 | 63 | rmo4 2957 |
. . 3
|
| 65 | 58, 64 | sylibr 134 |
. 2
|
| 66 | reu5 2714 |
. 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 709 wceq 1364 wfal 1369 wcel 2167 wral 2475 wrex 2476 wreu 2477 wrmo 2478 cin 3156 wss 3157 c0 3451 class class class wbr 4034
cc 7894
cr 7895
clt 8078
# cap 8625 |
| 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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-suploc 8017 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-br 4035 df-opab 4096 df-id 4329 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-iota 5220 df-fun 5261 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 |
| This theorem is referenced by: (None) |
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