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Mirrors > Home > MPE Home > Th. List > nnzi | Structured version Visualization version GIF version |
Description: A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
nnzi.1 | ⊢ 𝑁 ∈ ℕ |
Ref | Expression |
---|---|
nnzi | ⊢ 𝑁 ∈ ℤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnssz 11990 | . 2 ⊢ ℕ ⊆ ℤ | |
2 | nnzi.1 | . 2 ⊢ 𝑁 ∈ ℕ | |
3 | 1, 2 | sselii 3961 | 1 ⊢ 𝑁 ∈ ℤ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ℕcn 11626 ℤcz 11969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-i2m1 10593 ax-1ne0 10594 ax-rrecex 10597 ax-cnre 10598 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-neg 10861 df-nn 11627 df-z 11970 |
This theorem is referenced by: 1z 12000 2z 12002 3z 12003 4z 12004 faclbnd4lem1 13641 3dvds 15668 3dvdsdec 15669 divalglem6 15737 divalglem7 15738 divalglem8 15739 divalglem9 15740 ndvdsi 15751 6gcd4e2 15874 3lcm2e6 16060 prm23ge5 16140 pockthi 16231 modxai 16392 mod2xnegi 16395 gcdmodi 16398 strleun 16579 strle1 16580 lt6abl 18944 2logb9irr 25300 ppiublem1 25705 ppiublem2 25706 ppiub 25707 bpos1lem 25785 bposlem6 25792 bposlem8 25794 bposlem9 25795 lgsdir2lem5 25832 2lgsoddprmlem2 25912 ex-mod 28155 ex-dvds 28162 ex-gcd 28163 ex-lcm 28164 ballotlem1 31643 ballotlem2 31645 ballotlemfmpn 31651 ballotlemsdom 31668 ballotlemsel1i 31669 ballotlemsima 31672 ballotlemfrceq 31685 ballotlemfrcn0 31686 chtvalz 31799 hgt750lem 31821 inductionexd 40383 hoidmvlelem3 42756 fmtnoprmfac2lem1 43605 31prm 43637 mod42tp1mod8 43644 6even 43753 8even 43755 341fppr2 43776 8exp8mod9 43778 9fppr8 43779 nfermltl8rev 43784 nfermltlrev 43786 gbowge7 43805 gbege6 43807 stgoldbwt 43818 sbgoldbwt 43819 sbgoldbm 43826 mogoldbb 43827 sbgoldbo 43829 nnsum3primesle9 43836 nnsum4primeseven 43842 nnsum4primesevenALTV 43843 wtgoldbnnsum4prm 43844 bgoldbnnsum3prm 43846 bgoldbtbndlem1 43847 tgblthelfgott 43857 tgoldbach 43859 zlmodzxzequa 44479 zlmodzxznm 44480 zlmodzxzequap 44482 zlmodzxzldeplem3 44485 zlmodzxzldep 44487 ldepsnlinclem2 44489 ldepsnlinc 44491 |
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