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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 7992 | . . . . . 6 | |
2 | 1 | adantl 275 | . . . . 5 |
3 | suprzclex.ex | . . . . 5 | |
4 | 2, 3 | supclti 6973 | . . . 4 |
5 | 4 | ltm1d 8841 | . . 3 |
6 | suprzclex.ss | . . . . 5 | |
7 | zssre 9212 | . . . . 5 | |
8 | 6, 7 | sstrdi 3159 | . . . 4 |
9 | peano2rem 8179 | . . . . 5 | |
10 | 4, 9 | syl 14 | . . . 4 |
11 | 3, 8, 10 | suprlubex 8861 | . . 3 |
12 | 5, 11 | mpbid 146 | . 2 |
13 | 6 | adantr 274 | . . . . . . . . . 10 |
14 | 13 | sselda 3147 | . . . . . . . . 9 |
15 | 7, 14 | sselid 3145 | . . . . . . . 8 |
16 | 4 | adantr 274 | . . . . . . . . 9 |
17 | 16 | adantr 274 | . . . . . . . 8 |
18 | simprl 526 | . . . . . . . . . . . 12 | |
19 | 13, 18 | sseldd 3148 | . . . . . . . . . . 11 |
20 | zre 9209 | . . . . . . . . . . 11 | |
21 | 19, 20 | syl 14 | . . . . . . . . . 10 |
22 | peano2re 8048 | . . . . . . . . . 10 | |
23 | 21, 22 | syl 14 | . . . . . . . . 9 |
24 | 23 | adantr 274 | . . . . . . . 8 |
25 | 3 | ad2antrr 485 | . . . . . . . . 9 |
26 | 8 | ad2antrr 485 | . . . . . . . . 9 |
27 | simpr 109 | . . . . . . . . 9 | |
28 | 25, 26, 27 | suprubex 8860 | . . . . . . . 8 |
29 | simprr 527 | . . . . . . . . . 10 | |
30 | 1red 7928 | . . . . . . . . . . 11 | |
31 | 16, 30, 21 | ltsubaddd 8453 | . . . . . . . . . 10 |
32 | 29, 31 | mpbid 146 | . . . . . . . . 9 |
33 | 32 | adantr 274 | . . . . . . . 8 |
34 | 15, 17, 24, 28, 33 | lelttrd 8037 | . . . . . . 7 |
35 | 19 | adantr 274 | . . . . . . . 8 |
36 | zleltp1 9260 | . . . . . . . 8 | |
37 | 14, 35, 36 | syl2anc 409 | . . . . . . 7 |
38 | 34, 37 | mpbird 166 | . . . . . 6 |
39 | 38 | ralrimiva 2543 | . . . . 5 |
40 | breq2 3991 | . . . . . . . . . . . . 13 | |
41 | 40 | cbvrexv 2697 | . . . . . . . . . . . 12 |
42 | 41 | imbi2i 225 | . . . . . . . . . . 11 |
43 | 42 | ralbii 2476 | . . . . . . . . . 10 |
44 | 43 | anbi2i 454 | . . . . . . . . 9 |
45 | 44 | rexbii 2477 | . . . . . . . 8 |
46 | 3, 45 | sylib 121 | . . . . . . 7 |
47 | 46 | adantr 274 | . . . . . 6 |
48 | 13, 7 | sstrdi 3159 | . . . . . 6 |
49 | 47, 48, 21 | suprleubex 8863 | . . . . 5 |
50 | 39, 49 | mpbird 166 | . . . 4 |
51 | 47, 48, 18 | suprubex 8860 | . . . 4 |
52 | 16, 21 | letri3d 8028 | . . . 4 |
53 | 50, 51, 52 | mpbir2and 939 | . . 3 |
54 | 53, 18 | eqeltrd 2247 | . 2 |
55 | 12, 54 | rexlimddv 2592 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wral 2448 wrex 2449 wss 3121 class class class wbr 3987 (class class class)co 5851 csup 6957 cr 7766 c1 7768 caddc 7770 clt 7947 cle 7948 cmin 8083 cz 9205 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-addcom 7867 ax-addass 7869 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-0id 7875 ax-rnegex 7876 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-apti 7882 ax-pre-ltadd 7883 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-id 4276 df-po 4279 df-iso 4280 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-sup 6959 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-inn 8872 df-n0 9129 df-z 9206 |
This theorem is referenced by: infssuzcldc 11899 gcddvds 11911 |
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