Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 9335 | . 2 ⊢ (𝑦 ∈ (ℤ≥‘𝑀) → 𝑦 ∈ ℤ) | |
2 | 1 | ssriv 3101 | 1 ⊢ (ℤ≥‘𝑀) ⊆ ℤ |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 3071 ‘cfv 5123 ℤcz 9054 ℤ≥cuz 9326 |
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-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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-cnex 7711 ax-resscn 7712 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-neg 7936 df-z 9055 df-uz 9327 |
This theorem is referenced by: cau3 10887 climz 11061 serclim0 11074 climaddc1 11098 climmulc2 11100 climsubc1 11101 climsubc2 11102 climle 11103 climlec2 11110 summodclem2a 11150 summodclem2 11151 zsumdc 11153 fsum3cvg3 11165 iserabs 11244 isumshft 11259 explecnv 11274 clim2prod 11308 prodfclim1 11313 ntrivcvgap 11317 prodmodclem2a 11345 prodmodclem2 11346 infssuzcldc 11644 exmidunben 11939 lmbrf 12384 lmres 12417 climcncf 12740 |
Copyright terms: Public domain | W3C validator |