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| Mirrors > Home > MPE Home > Th. List > uzssz | Structured version Visualization version 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 | uzf 12881 | . . . . 5 ⊢ ℤ≥:ℤ⟶𝒫 ℤ | |
| 2 | 1 | ffvelcdmi 7103 | . . . 4 ⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) ∈ 𝒫 ℤ) |
| 3 | 2 | elpwid 4609 | . . 3 ⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) ⊆ ℤ) |
| 4 | 1 | fdmi 6747 | . . 3 ⊢ dom ℤ≥ = ℤ |
| 5 | 3, 4 | eleq2s 2859 | . 2 ⊢ (𝑀 ∈ dom ℤ≥ → (ℤ≥‘𝑀) ⊆ ℤ) |
| 6 | ndmfv 6941 | . . 3 ⊢ (¬ 𝑀 ∈ dom ℤ≥ → (ℤ≥‘𝑀) = ∅) | |
| 7 | 0ss 4400 | . . 3 ⊢ ∅ ⊆ ℤ | |
| 8 | 6, 7 | eqsstrdi 4028 | . 2 ⊢ (¬ 𝑀 ∈ dom ℤ≥ → (ℤ≥‘𝑀) ⊆ ℤ) |
| 9 | 5, 8 | pm2.61i 182 | 1 ⊢ (ℤ≥‘𝑀) ⊆ ℤ |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∈ wcel 2108 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 dom cdm 5685 ‘cfv 6561 ℤcz 12613 ℤ≥cuz 12878 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-neg 11495 df-z 12614 df-uz 12879 |
| This theorem is referenced by: uzssre 12900 uzwo 12953 uzwo2 12954 infssuzle 12973 infssuzcl 12974 uzsupss 12982 uzwo3 12985 uzsup 13903 cau3 15394 caubnd 15397 limsupgre 15517 rlimclim 15582 climz 15585 climaddc1 15671 climmulc2 15673 climsubc1 15674 climsubc2 15675 climlec2 15695 isercolllem1 15701 isercolllem2 15702 isercoll 15704 caurcvg 15713 caucvg 15715 iseraltlem1 15718 iseraltlem2 15719 iseraltlem3 15720 summolem2a 15751 summolem2 15752 zsum 15754 fsumcvg3 15765 climfsum 15856 divcnvshft 15891 clim2prod 15924 ntrivcvg 15933 ntrivcvgfvn0 15935 ntrivcvgtail 15936 ntrivcvgmullem 15937 ntrivcvgmul 15938 prodrblem 15965 prodmolem2a 15970 prodmolem2 15971 zprod 15973 4sqlem11 16993 gsumval3 19925 lmbrf 23268 lmres 23308 uzrest 23905 uzfbas 23906 lmflf 24013 lmmbrf 25296 iscau4 25313 iscauf 25314 caucfil 25317 lmclimf 25338 mbfsup 25699 mbflimsup 25701 ig1pdvds 26219 ulmval 26423 ulmpm 26426 2sqlem6 27467 ballotlemfc0 34495 ballotlemfcc 34496 ballotlemiex 34504 ballotlemsima 34518 ballotlemrv2 34524 breprexplemc 34647 erdszelem4 35199 erdszelem8 35203 caures 37767 diophin 42783 irrapxlem1 42833 monotuz 42953 hashnzfzclim 44341 uzmptshftfval 44365 uzct 45068 uzfissfz 45337 ssuzfz 45360 uzssre2 45418 uzssz2 45467 uzinico2 45575 fnlimfvre 45689 climleltrp 45691 limsupequzmpt2 45733 limsupequzlem 45737 liminfequzmpt2 45806 ioodvbdlimc1lem2 45947 ioodvbdlimc2lem 45949 sge0isum 46442 smflimlem1 46786 smflimlem2 46787 smflim 46792 |
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