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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 12937 | . 2 ⊢ (𝑎 ∈ (1...𝐴) → 𝑎 ∈ ℕ) | |
2 | 1 | ssriv 3971 | 1 ⊢ (1...𝐴) ⊆ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3936 (class class class)co 7156 1c1 10538 ℕcn 11638 ...cfz 12893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-z 11983 df-uz 12245 df-fz 12894 |
This theorem is referenced by: fzssnn 12952 fzossnn 13087 isercoll 15024 prmreclem2 16253 prmreclem3 16254 vdwnnlem1 16331 prmodvdslcmf 16383 gsumval3 19027 1stcfb 22053 1stckgenlem 22161 ovoliunlem1 24103 ovoliun2 24107 ovolicc2lem4 24121 uniioovol 24180 uniioombllem4 24187 lgamgulm2 25613 lgamcvglem 25617 fsumvma2 25790 dchrmusum2 26070 dchrvmasum2lem 26072 mudivsum 26106 mulogsum 26108 mulog2sumlem2 26111 padct 30455 psgnfzto1stlem 30742 fzto1st1 30744 smatrcl 31061 smatlem 31062 smattr 31064 smatbl 31065 smatbr 31066 1smat1 31069 submateqlem1 31072 submateqlem2 31073 submateq 31074 madjusmdetlem2 31093 madjusmdetlem3 31094 madjusmdetlem4 31095 mdetlap 31097 esumsup 31348 esumgect 31349 carsggect 31576 carsgclctunlem2 31577 ballotlemsup 31762 fsum2dsub 31878 reprgt 31892 reprfi2 31894 reprfz1 31895 hashrepr 31896 breprexplema 31901 breprexplemc 31903 breprexp 31904 breprexpnat 31905 vtscl 31909 circlemeth 31911 hgt750lemd 31919 hgt750lemb 31927 hgt750leme 31929 eldioph4b 39428 diophren 39430 caratheodorylem2 42829 hoidmvlelem2 42898 |
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