Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7837 | . . . . . 6 | |
2 | 1 | adantl 275 | . . . . 5 |
3 | suprzclex.ex | . . . . 5 | |
4 | 2, 3 | supclti 6878 | . . . 4 |
5 | 4 | ltm1d 8683 | . . 3 |
6 | suprzclex.ss | . . . . 5 | |
7 | zssre 9054 | . . . . 5 | |
8 | 6, 7 | sstrdi 3104 | . . . 4 |
9 | peano2rem 8022 | . . . . 5 | |
10 | 4, 9 | syl 14 | . . . 4 |
11 | 3, 8, 10 | suprlubex 8703 | . . 3 |
12 | 5, 11 | mpbid 146 | . 2 |
13 | 6 | adantr 274 | . . . . . . . . . 10 |
14 | 13 | sselda 3092 | . . . . . . . . 9 |
15 | 7, 14 | sseldi 3090 | . . . . . . . 8 |
16 | 4 | adantr 274 | . . . . . . . . 9 |
17 | 16 | adantr 274 | . . . . . . . 8 |
18 | simprl 520 | . . . . . . . . . . . 12 | |
19 | 13, 18 | sseldd 3093 | . . . . . . . . . . 11 |
20 | zre 9051 | . . . . . . . . . . 11 | |
21 | 19, 20 | syl 14 | . . . . . . . . . 10 |
22 | peano2re 7891 | . . . . . . . . . 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 8702 | . . . . . . . 8 |
29 | simprr 521 | . . . . . . . . . 10 | |
30 | 1red 7774 | . . . . . . . . . . 11 | |
31 | 16, 30, 21 | ltsubaddd 8296 | . . . . . . . . . 10 |
32 | 29, 31 | mpbid 146 | . . . . . . . . 9 |
33 | 32 | adantr 274 | . . . . . . . 8 |
34 | 15, 17, 24, 28, 33 | lelttrd 7880 | . . . . . . 7 |
35 | 19 | adantr 274 | . . . . . . . 8 |
36 | zleltp1 9102 | . . . . . . . 8 | |
37 | 14, 35, 36 | syl2anc 408 | . . . . . . 7 |
38 | 34, 37 | mpbird 166 | . . . . . 6 |
39 | 38 | ralrimiva 2503 | . . . . 5 |
40 | breq2 3928 | . . . . . . . . . . . . 13 | |
41 | 40 | cbvrexv 2653 | . . . . . . . . . . . 12 |
42 | 41 | imbi2i 225 | . . . . . . . . . . 11 |
43 | 42 | ralbii 2439 | . . . . . . . . . 10 |
44 | 43 | anbi2i 452 | . . . . . . . . 9 |
45 | 44 | rexbii 2440 | . . . . . . . 8 |
46 | 3, 45 | sylib 121 | . . . . . . 7 |
47 | 46 | adantr 274 | . . . . . 6 |
48 | 13, 7 | sstrdi 3104 | . . . . . 6 |
49 | 47, 48, 21 | suprleubex 8705 | . . . . 5 |
50 | 39, 49 | mpbird 166 | . . . 4 |
51 | 47, 48, 18 | suprubex 8702 | . . . 4 |
52 | 16, 21 | letri3d 7872 | . . . 4 |
53 | 50, 51, 52 | mpbir2and 928 | . . 3 |
54 | 53, 18 | eqeltrd 2214 | . 2 |
55 | 12, 54 | rexlimddv 2552 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2414 wrex 2415 wss 3066 class class class wbr 3924 (class class class)co 5767 csup 6862 cr 7612 c1 7614 caddc 7616 clt 7793 cle 7794 cmin 7926 cz 9047 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-id 4210 df-po 4213 df-iso 4214 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-sup 6864 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 |
This theorem is referenced by: infssuzcldc 11633 gcddvds 11641 |
Copyright terms: Public domain | W3C validator |