![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 12563 | . 2 ⊢ ℕ0 ⊆ ℤ | |
2 | nn0zi.1 | . 2 ⊢ 𝑁 ∈ ℕ0 | |
3 | 1, 2 | sselii 3975 | 1 ⊢ 𝑁 ∈ ℤ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 ℕ0cn0 12454 ℤcz 12540 |
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-pr 5420 ax-un 7708 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-i2m1 11160 ax-1ne0 11161 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 |
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-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-ov 7396 df-om 7839 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-neg 11429 df-nn 12195 df-n0 12455 df-z 12541 |
This theorem is referenced by: le9lt10 12686 expnass 14154 faclbnd4lem1 14235 efsep 16035 3dvdsdec 16257 3dvds2dec 16258 divalglem0 16318 divalglem2 16320 ndvdsi 16337 gcdaddmlem 16447 6lcm4e12 16535 phicl2 16683 dec2dvds 16978 dec5dvds2 16980 modxai 16983 mod2xnegi 16986 gcdi 16988 gcdmodi 16989 1259lem1 17046 1259lem2 17047 1259lem3 17048 1259lem4 17049 1259lem5 17050 2503lem1 17052 2503lem2 17053 2503lem3 17054 4001lem1 17056 4001lem2 17057 4001lem3 17058 4001lem4 17059 ppi1i 26599 ppi2i 26600 ppiublem1 26632 konigsberglem5 29374 dp2lt10 31921 dp2ltc 31924 ballotlemfelz 33320 hgt750lemd 33491 hgt750lem 33494 hgt750leme 33501 poimirlem26 36318 poimirlem28 36320 fmtno4prmfac 46012 31prm 46037 nfermltl2rev 46183 linevalexample 46724 ackval42 47030 |
Copyright terms: Public domain | W3C validator |