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| Mirrors > Home > MPE Home > Th. List > fz1ssnn | Structured version Visualization version GIF version | ||
| Description: A finite set of positive integers is a set of positive integers. (Contributed by Stefan O'Rear, 16-Oct-2014.) | 
| Ref | Expression | 
|---|---|
| fz1ssnn | ⊢ (1...𝐴) ⊆ ℕ | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elfznn 13594 | . 2 ⊢ (𝑎 ∈ (1...𝐴) → 𝑎 ∈ ℕ) | |
| 2 | 1 | ssriv 3986 | 1 ⊢ (1...𝐴) ⊆ ℕ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ⊆ wss 3950 (class class class)co 7432 1c1 11157 ℕcn 12267 ...cfz 13548 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-z 12616 df-uz 12880 df-fz 13549 | 
| This theorem is referenced by: fzssnn 13609 fzossnn 13752 isercoll 15705 prmreclem2 16956 prmreclem3 16957 vdwnnlem1 17034 prmodvdslcmf 17086 gsumval3 19926 1stcfb 23454 1stckgenlem 23562 ovoliunlem1 25538 ovoliun2 25542 ovolicc2lem4 25556 uniioovol 25615 uniioombllem4 25622 lgamgulm2 27080 lgamcvglem 27084 fsumvma2 27259 dchrmusum2 27539 dchrvmasum2lem 27541 mudivsum 27575 mulogsum 27577 mulog2sumlem2 27580 padct 32732 psgnfzto1stlem 33121 fzto1st1 33123 smatrcl 33796 smatlem 33797 smattr 33799 smatbl 33800 smatbr 33801 1smat1 33804 submateqlem1 33807 submateqlem2 33808 submateq 33809 madjusmdetlem2 33828 madjusmdetlem3 33829 madjusmdetlem4 33830 mdetlap 33832 esumsup 34091 esumgect 34092 carsggect 34321 carsgclctunlem2 34322 ballotlemsup 34508 fsum2dsub 34623 reprgt 34637 reprfi2 34639 reprfz1 34640 hashrepr 34641 breprexplema 34646 breprexplemc 34648 breprexp 34649 breprexpnat 34650 vtscl 34654 circlemeth 34656 hgt750lemd 34664 hgt750lemb 34672 hgt750leme 34674 lcmineqlem4 42034 lcmineqlem6 42036 lcmineqlem15 42045 lcmineqlem16 42046 lcmineqlem19 42049 lcmineqlem20 42050 lcmineqlem21 42051 lcmineqlem22 42052 sticksstones1 42148 fisdomnn 42285 sumcubes 42352 eldioph4b 42827 diophren 42829 caratheodorylem2 46547 hoidmvlelem2 46616 | 
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