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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 7579 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 15093 |
. 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 15094 |
. . . . . . . 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 15094 |
. . . . . . . 8
#
|
| 44 | simpr 110 |
. . . . . . . . 9
# # | |
| 45 | reaplt 8661 |
. . . . . . . . . 10
#
| |
| 46 | 19, 24, 45 | syl2anc 411 |
. . . . . . . . 9
# #
|
| 47 | 44, 46 | mpbid 147 |
. . . . . . . 8
# |
| 48 | 29, 43, 47 | mpjaodan 800 |
. . . . . . 7
# |
| 49 | 48 | inegd 1392 |
. . . . . 6
# |
| 50 | simplrl 535 |
. . . . . . . 8
| |
| 51 | 50 | recnd 8101 |
. . . . . . 7
 |
| 52 | simplrr 536 |
. . . . . . . 8
| |
| 53 | 52 | recnd 8101 |
. . . . . . 7
|
| 54 | apti 8695 |
. . . . . . 7
# | |
| 55 | 51, 53, 54 | syl2anc 411 |
. . . . . 6
# |
| 56 | 49, 55 | mpbird 167 |
. . . . 5
|
| 57 | 56 | ex 115 |
. . . 4
|
| 58 | 57 | ralrimivva 2588 |
. . 3
|
| 59 | breq2 4048 |
. . . . . 6
| |
| 60 | 59 | ralbidv 2506 |
. . . . 5
|
| 61 | breq1 4047 |
. . . . . 6
| |
| 62 | 61 | ralbidv 2506 |
. . . . 5
|
| 63 | 60, 62 | anbi12d 473 |
. . . 4
|
| 64 | 63 | rmo4 2966 |
. . 3
|
| 65 | 58, 64 | sylibr 134 |
. 2
|
| 66 | reu5 2723 |
. 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 710 wceq 1373 wfal 1378 wcel 2176 wral 2484 wrex 2485 wreu 2486 wrmo 2487 cin 3165 wss 3166 c0 3460 class class class wbr 4044
cc 7923
cr 7924
clt 8107
# cap 8654 |
| 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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 ax-pre-mulgt0 8042 ax-pre-suploc 8046 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-br 4045 df-opab 4106 df-id 4340 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-iota 5232 df-fun 5273 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-reap 8648 df-ap 8655 |
| This theorem is referenced by: (None) |
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