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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 13512 | . 2 ⊢ (𝑎 ∈ (1...𝐴) → 𝑎 ∈ ℕ) | |
2 | 1 | ssriv 3982 | 1 ⊢ (1...𝐴) ⊆ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3944 (class class class)co 7393 1c1 11093 ℕcn 12194 ...cfz 13466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-z 12541 df-uz 12805 df-fz 13467 |
This theorem is referenced by: fzssnn 13527 fzossnn 13663 isercoll 15596 prmreclem2 16832 prmreclem3 16833 vdwnnlem1 16910 prmodvdslcmf 16962 gsumval3 19734 1stcfb 22878 1stckgenlem 22986 ovoliunlem1 24948 ovoliun2 24952 ovolicc2lem4 24966 uniioovol 25025 uniioombllem4 25032 lgamgulm2 26467 lgamcvglem 26471 fsumvma2 26644 dchrmusum2 26924 dchrvmasum2lem 26926 mudivsum 26960 mulogsum 26962 mulog2sumlem2 26965 padct 31815 psgnfzto1stlem 32130 fzto1st1 32132 smatrcl 32605 smatlem 32606 smattr 32608 smatbl 32609 smatbr 32610 1smat1 32613 submateqlem1 32616 submateqlem2 32617 submateq 32618 madjusmdetlem2 32637 madjusmdetlem3 32638 madjusmdetlem4 32639 mdetlap 32641 esumsup 32916 esumgect 32917 carsggect 33146 carsgclctunlem2 33147 ballotlemsup 33332 fsum2dsub 33448 reprgt 33462 reprfi2 33464 reprfz1 33465 hashrepr 33466 breprexplema 33471 breprexplemc 33473 breprexp 33474 breprexpnat 33475 vtscl 33479 circlemeth 33481 hgt750lemd 33489 hgt750lemb 33497 hgt750leme 33499 lcmineqlem4 40700 lcmineqlem6 40702 lcmineqlem15 40711 lcmineqlem16 40712 lcmineqlem19 40715 lcmineqlem20 40716 lcmineqlem21 40717 lcmineqlem22 40718 sticksstones1 40765 eldioph4b 41318 diophren 41320 caratheodorylem2 45014 hoidmvlelem2 45083 |
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