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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 11728 | . . . . 5 ⊢ ℤ≥:ℤ⟶𝒫 ℤ | |
2 | 1 | ffvelrni 6398 | . . . 4 ⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) ∈ 𝒫 ℤ) |
3 | 2 | elpwid 4203 | . . 3 ⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) ⊆ ℤ) |
4 | 1 | fdmi 6090 | . . 3 ⊢ dom ℤ≥ = ℤ |
5 | 3, 4 | eleq2s 2748 | . 2 ⊢ (𝑀 ∈ dom ℤ≥ → (ℤ≥‘𝑀) ⊆ ℤ) |
6 | ndmfv 6256 | . . 3 ⊢ (¬ 𝑀 ∈ dom ℤ≥ → (ℤ≥‘𝑀) = ∅) | |
7 | 0ss 4005 | . . 3 ⊢ ∅ ⊆ ℤ | |
8 | 6, 7 | syl6eqss 3688 | . 2 ⊢ (¬ 𝑀 ∈ dom ℤ≥ → (ℤ≥‘𝑀) ⊆ ℤ) |
9 | 5, 8 | pm2.61i 176 | 1 ⊢ (ℤ≥‘𝑀) ⊆ ℤ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2030 ⊆ wss 3607 ∅c0 3948 𝒫 cpw 4191 dom cdm 5143 ‘cfv 5926 ℤcz 11415 ℤ≥cuz 11725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-cnex 10030 ax-resscn 10031 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-fv 5934 df-ov 6693 df-neg 10307 df-z 11416 df-uz 11726 |
This theorem is referenced by: uzwo 11789 uzwo2 11790 infssuzle 11809 infssuzcl 11810 uzsupss 11818 uzwo3 11821 fzof 12506 uzsup 12702 cau3 14139 caubnd 14142 limsupgre 14256 rlimclim 14321 climz 14324 climaddc1 14409 climmulc2 14411 climsubc1 14412 climsubc2 14413 climlec2 14433 isercolllem1 14439 isercolllem2 14440 isercoll 14442 caurcvg 14451 caucvg 14453 iseraltlem1 14456 iseraltlem2 14457 iseraltlem3 14458 summolem2a 14490 summolem2 14491 zsum 14493 fsumcvg3 14504 climfsum 14596 divcnvshft 14631 clim2prod 14664 ntrivcvg 14673 ntrivcvgfvn0 14675 ntrivcvgtail 14676 ntrivcvgmullem 14677 ntrivcvgmul 14678 prodrblem 14703 prodmolem2a 14708 prodmolem2 14709 zprod 14711 4sqlem11 15706 gsumval3 18354 lmbrf 21112 lmres 21152 uzrest 21748 uzfbas 21749 lmflf 21856 lmmbrf 23106 iscau4 23123 iscauf 23124 caucfil 23127 lmclimf 23148 mbfsup 23476 mbflimsup 23478 ig1pdvds 23981 ulmval 24179 ulmpm 24182 2sqlem6 25193 ballotlemfc0 30682 ballotlemfcc 30683 ballotlemiex 30691 ballotlemsdom 30701 ballotlemsima 30705 ballotlemrv2 30711 breprexplemc 30838 erdszelem4 31302 erdszelem8 31306 caures 33686 diophin 37653 irrapxlem1 37703 monotuz 37823 hashnzfzclim 38838 uzmptshftfval 38862 uzct 39546 uzfissfz 39855 ssuzfz 39878 uzssre 39933 uzssre2 39946 uzssz2 39998 uzinico2 40107 fnlimfvre 40224 climleltrp 40226 limsupequzmpt2 40268 limsupequzlem 40272 liminfequzmpt2 40341 ioodvbdlimc1lem2 40465 ioodvbdlimc2lem 40467 sge0isum 40962 smflimlem1 41300 smflimlem2 41301 smflim 41306 |
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