![]() |
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 9359 | . 2 ⊢ (𝑦 ∈ (ℤ≥‘𝑀) → 𝑦 ∈ ℤ) | |
2 | 1 | ssriv 3106 | 1 ⊢ (ℤ≥‘𝑀) ⊆ ℤ |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 3076 ‘cfv 5131 ℤcz 9078 ℤ≥cuz 9350 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-cnex 7735 ax-resscn 7736 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-ov 5785 df-neg 7960 df-z 9079 df-uz 9351 |
This theorem is referenced by: cau3 10919 climz 11093 serclim0 11106 climaddc1 11130 climmulc2 11132 climsubc1 11133 climsubc2 11134 climle 11135 climlec2 11142 summodclem2a 11182 summodclem2 11183 zsumdc 11185 fsum3cvg3 11197 iserabs 11276 isumshft 11291 explecnv 11306 clim2prod 11340 prodfclim1 11345 ntrivcvgap 11349 prodmodclem2a 11377 prodmodclem2 11378 zproddc 11380 infssuzcldc 11680 exmidunben 11975 lmbrf 12423 lmres 12456 climcncf 12779 |
Copyright terms: Public domain | W3C validator |