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Mirrors > Home > ILE Home > Th. List > suprzubdc | Unicode version |
Description: The supremum of a bounded-above decidable set of integers is greater than any member of the set. (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 5-Oct-2024.) |
Ref | Expression |
---|---|
suprzubdc.ss | |
suprzubdc.dc | DECID |
suprzubdc.ub | |
suprzubdc.b |
Ref | Expression |
---|---|
suprzubdc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suprzubdc.ub | . . 3 | |
2 | breq2 3971 | . . . . 5 | |
3 | 2 | ralbidv 2457 | . . . 4 |
4 | 3 | cbvrexv 2681 | . . 3 |
5 | 1, 4 | sylib 121 | . 2 |
6 | dfin5 3109 | . . . . . . 7 | |
7 | suprzubdc.ss | . . . . . . . 8 | |
8 | sseqin2 3327 | . . . . . . . 8 | |
9 | 7, 8 | sylib 121 | . . . . . . 7 |
10 | 6, 9 | eqtr3id 2204 | . . . . . 6 |
11 | 10 | supeq1d 6934 | . . . . 5 |
12 | 11 | adantr 274 | . . . 4 |
13 | suprzubdc.b | . . . . . . 7 | |
14 | 7, 13 | sseldd 3129 | . . . . . 6 |
15 | 14 | adantr 274 | . . . . 5 |
16 | eleq1 2220 | . . . . 5 | |
17 | 13 | adantr 274 | . . . . 5 |
18 | eleq1w 2218 | . . . . . . 7 | |
19 | 18 | dcbid 824 | . . . . . 6 DECID DECID |
20 | suprzubdc.dc | . . . . . . 7 DECID | |
21 | 20 | ad2antrr 480 | . . . . . 6 DECID |
22 | eluzelz 9454 | . . . . . . 7 | |
23 | 22 | adantl 275 | . . . . . 6 |
24 | 19, 21, 23 | rspcdva 2821 | . . . . 5 DECID |
25 | simprl 521 | . . . . . . . 8 | |
26 | 25 | peano2zd 9295 | . . . . . . 7 |
27 | 15 | zred 9292 | . . . . . . . 8 |
28 | 25 | zred 9292 | . . . . . . . 8 |
29 | 26 | zred 9292 | . . . . . . . 8 |
30 | breq1 3970 | . . . . . . . . 9 | |
31 | simprr 522 | . . . . . . . . 9 | |
32 | 30, 31, 17 | rspcdva 2821 | . . . . . . . 8 |
33 | 28 | lep1d 8808 | . . . . . . . 8 |
34 | 27, 28, 29, 32, 33 | letrd 8004 | . . . . . . 7 |
35 | eluz2 9451 | . . . . . . 7 | |
36 | 15, 26, 34, 35 | syl3anbrc 1166 | . . . . . 6 |
37 | eluzle 9457 | . . . . . . . . . 10 | |
38 | 37 | ad2antlr 481 | . . . . . . . . 9 |
39 | 25 | ad2antrr 480 | . . . . . . . . . 10 |
40 | eluzelz 9454 | . . . . . . . . . . 11 | |
41 | 40 | ad2antlr 481 | . . . . . . . . . 10 |
42 | zltp1le 9227 | . . . . . . . . . 10 | |
43 | 39, 41, 42 | syl2anc 409 | . . . . . . . . 9 |
44 | 38, 43 | mpbird 166 | . . . . . . . 8 |
45 | 41 | zred 9292 | . . . . . . . . 9 |
46 | 28 | ad2antrr 480 | . . . . . . . . 9 |
47 | breq1 3970 | . . . . . . . . . 10 | |
48 | 31 | ad2antrr 480 | . . . . . . . . . 10 |
49 | simpr 109 | . . . . . . . . . 10 | |
50 | 47, 48, 49 | rspcdva 2821 | . . . . . . . . 9 |
51 | 45, 46, 50 | lensymd 8002 | . . . . . . . 8 |
52 | 44, 51 | pm2.65da 651 | . . . . . . 7 |
53 | 52 | ralrimiva 2530 | . . . . . 6 |
54 | fveq2 5471 | . . . . . . . 8 | |
55 | 54 | raleqdv 2658 | . . . . . . 7 |
56 | 55 | rspcev 2816 | . . . . . 6 |
57 | 36, 53, 56 | syl2anc 409 | . . . . 5 |
58 | 15, 16, 17, 24, 57 | zsupcl 11847 | . . . 4 |
59 | 12, 58 | eqeltrrd 2235 | . . 3 |
60 | eluzle 9457 | . . 3 | |
61 | 59, 60 | syl 14 | . 2 |
62 | 5, 61 | rexlimddv 2579 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 DECID wdc 820 wceq 1335 wcel 2128 wral 2435 wrex 2436 crab 2439 cin 3101 wss 3102 class class class wbr 3967 cfv 5173 (class class class)co 5827 csup 6929 cr 7734 c1 7736 caddc 7738 clt 7915 cle 7916 cz 9173 cuz 9445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 ax-cnex 7826 ax-resscn 7827 ax-1cn 7828 ax-1re 7829 ax-icn 7830 ax-addcl 7831 ax-addrcl 7832 ax-mulcl 7833 ax-addcom 7835 ax-addass 7837 ax-distr 7839 ax-i2m1 7840 ax-0lt1 7841 ax-0id 7843 ax-rnegex 7844 ax-cnre 7846 ax-pre-ltirr 7847 ax-pre-ltwlin 7848 ax-pre-lttrn 7849 ax-pre-apti 7850 ax-pre-ltadd 7851 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4029 df-mpt 4030 df-id 4256 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-fv 5181 df-riota 5783 df-ov 5830 df-oprab 5831 df-mpo 5832 df-1st 6091 df-2nd 6092 df-sup 6931 df-pnf 7917 df-mnf 7918 df-xr 7919 df-ltxr 7920 df-le 7921 df-sub 8053 df-neg 8054 df-inn 8840 df-n0 9097 df-z 9174 df-uz 9446 df-fz 9920 df-fzo 10052 |
This theorem is referenced by: pcprendvds 12181 pcpremul 12184 |
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