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Mirrors > Home > ILE Home > Th. List > suprzclex | Unicode version |
Description: The supremum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 20-Dec-2021.) |
Ref | Expression |
---|---|
suprzclex.ex | |
suprzclex.ss |
Ref | Expression |
---|---|
suprzclex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lttri3 7812 | . . . . . 6 | |
2 | 1 | adantl 275 | . . . . 5 |
3 | suprzclex.ex | . . . . 5 | |
4 | 2, 3 | supclti 6853 | . . . 4 |
5 | 4 | ltm1d 8658 | . . 3 |
6 | suprzclex.ss | . . . . 5 | |
7 | zssre 9029 | . . . . 5 | |
8 | 6, 7 | sstrdi 3079 | . . . 4 |
9 | peano2rem 7997 | . . . . 5 | |
10 | 4, 9 | syl 14 | . . . 4 |
11 | 3, 8, 10 | suprlubex 8678 | . . 3 |
12 | 5, 11 | mpbid 146 | . 2 |
13 | 6 | adantr 274 | . . . . . . . . . 10 |
14 | 13 | sselda 3067 | . . . . . . . . 9 |
15 | 7, 14 | sseldi 3065 | . . . . . . . 8 |
16 | 4 | adantr 274 | . . . . . . . . 9 |
17 | 16 | adantr 274 | . . . . . . . 8 |
18 | simprl 505 | . . . . . . . . . . . 12 | |
19 | 13, 18 | sseldd 3068 | . . . . . . . . . . 11 |
20 | zre 9026 | . . . . . . . . . . 11 | |
21 | 19, 20 | syl 14 | . . . . . . . . . 10 |
22 | peano2re 7866 | . . . . . . . . . 10 | |
23 | 21, 22 | syl 14 | . . . . . . . . 9 |
24 | 23 | adantr 274 | . . . . . . . 8 |
25 | 3 | ad2antrr 479 | . . . . . . . . 9 |
26 | 8 | ad2antrr 479 | . . . . . . . . 9 |
27 | simpr 109 | . . . . . . . . 9 | |
28 | 25, 26, 27 | suprubex 8677 | . . . . . . . 8 |
29 | simprr 506 | . . . . . . . . . 10 | |
30 | 1red 7749 | . . . . . . . . . . 11 | |
31 | 16, 30, 21 | ltsubaddd 8271 | . . . . . . . . . 10 |
32 | 29, 31 | mpbid 146 | . . . . . . . . 9 |
33 | 32 | adantr 274 | . . . . . . . 8 |
34 | 15, 17, 24, 28, 33 | lelttrd 7855 | . . . . . . 7 |
35 | 19 | adantr 274 | . . . . . . . 8 |
36 | zleltp1 9077 | . . . . . . . 8 | |
37 | 14, 35, 36 | syl2anc 408 | . . . . . . 7 |
38 | 34, 37 | mpbird 166 | . . . . . 6 |
39 | 38 | ralrimiva 2482 | . . . . 5 |
40 | breq2 3903 | . . . . . . . . . . . . 13 | |
41 | 40 | cbvrexv 2632 | . . . . . . . . . . . 12 |
42 | 41 | imbi2i 225 | . . . . . . . . . . 11 |
43 | 42 | ralbii 2418 | . . . . . . . . . 10 |
44 | 43 | anbi2i 452 | . . . . . . . . 9 |
45 | 44 | rexbii 2419 | . . . . . . . 8 |
46 | 3, 45 | sylib 121 | . . . . . . 7 |
47 | 46 | adantr 274 | . . . . . 6 |
48 | 13, 7 | sstrdi 3079 | . . . . . 6 |
49 | 47, 48, 21 | suprleubex 8680 | . . . . 5 |
50 | 39, 49 | mpbird 166 | . . . 4 |
51 | 47, 48, 18 | suprubex 8677 | . . . 4 |
52 | 16, 21 | letri3d 7847 | . . . 4 |
53 | 50, 51, 52 | mpbir2and 913 | . . 3 |
54 | 53, 18 | eqeltrd 2194 | . 2 |
55 | 12, 54 | rexlimddv 2531 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1316 wcel 1465 wral 2393 wrex 2394 wss 3041 class class class wbr 3899 (class class class)co 5742 csup 6837 cr 7587 c1 7589 caddc 7591 clt 7768 cle 7769 cmin 7901 cz 9022 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-sup 6839 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-n0 8946 df-z 9023 |
This theorem is referenced by: infssuzcldc 11571 gcddvds 11579 |
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