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Mirrors > Home > ILE Home > Th. List > uzssz | GIF version |
Description: An upper set of integers is a subset of all integers. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.) |
Ref | Expression |
---|---|
uzssz | ⊢ (ℤ≥‘𝑀) ⊆ ℤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz 9089 | . 2 ⊢ (𝑦 ∈ (ℤ≥‘𝑀) → 𝑦 ∈ ℤ) | |
2 | 1 | ssriv 3030 | 1 ⊢ (ℤ≥‘𝑀) ⊆ ℤ |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 3000 ‘cfv 5028 ℤcz 8811 ℤ≥cuz 9080 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-cnex 7497 ax-resscn 7498 |
This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-rab 2369 df-v 2622 df-sbc 2842 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-mpt 3907 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-fv 5036 df-ov 5669 df-neg 7717 df-z 8812 df-uz 9081 |
This theorem is referenced by: cau3 10609 climz 10741 serclim0 10754 iserclim0 10755 climaddc1 10778 climmulc2 10780 climsubc1 10781 climsubc2 10782 climle 10783 climlec2 10791 isummolem2a 10832 isummolem2 10833 zisum 10835 fsum3cvg3 10850 iserabs 10930 isumshft 10945 explecnv 10960 infssuzcldc 11286 climcncf 11913 |
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