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Mirrors > Home > MPE Home > Th. List > nn0zi | Structured version Visualization version GIF version |
Description: A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
nn0zi.1 | ⊢ 𝑁 ∈ ℕ0 |
Ref | Expression |
---|---|
nn0zi | ⊢ 𝑁 ∈ ℤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0ssz 11991 | . 2 ⊢ ℕ0 ⊆ ℤ | |
2 | nn0zi.1 | . 2 ⊢ 𝑁 ∈ ℕ0 | |
3 | 1, 2 | sselii 3912 | 1 ⊢ 𝑁 ∈ ℤ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 ℕ0cn0 11885 ℤcz 11969 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-i2m1 10594 ax-1ne0 10595 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 |
This theorem is referenced by: le9lt10 12113 expnass 13566 faclbnd4lem1 13649 efsep 15455 3dvdsdec 15673 3dvds2dec 15674 divalglem0 15734 divalglem2 15736 ndvdsi 15753 gcdaddmlem 15862 6lcm4e12 15950 phicl2 16095 dec2dvds 16389 dec5dvds2 16391 modxai 16394 mod2xnegi 16397 gcdi 16399 gcdmodi 16400 1259lem1 16456 1259lem2 16457 1259lem3 16458 1259lem4 16459 1259lem5 16460 2503lem1 16462 2503lem2 16463 2503lem3 16464 4001lem1 16466 4001lem2 16467 4001lem3 16468 4001lem4 16469 ppi1i 25753 ppi2i 25754 ppiublem1 25786 konigsberglem5 28041 dp2lt10 30586 dp2ltc 30589 ballotlemfelz 31858 hgt750lemd 32029 hgt750lem 32032 hgt750leme 32039 poimirlem26 35083 poimirlem28 35085 3lexlogpow5ineq1 39341 fmtno4prmfac 44089 31prm 44114 nfermltl2rev 44261 linevalexample 44804 ackval42 45110 |
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