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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 12636 | . 2 ⊢ ℕ0 ⊆ ℤ | |
| 2 | nn0zi.1 | . 2 ⊢ 𝑁 ∈ ℕ0 | |
| 3 | 1, 2 | sselii 3980 | 1 ⊢ 𝑁 ∈ ℤ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ℕ0cn0 12526 ℤcz 12613 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-i2m1 11223 ax-1ne0 11224 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 |
| This theorem is referenced by: le9lt10 12760 fz0to5un2tp 13671 expnass 14247 faclbnd4lem1 14332 efsep 16146 3dvdsdec 16369 3dvds2dec 16370 divalglem0 16430 divalglem2 16432 ndvdsi 16449 gcdaddmlem 16561 6lcm4e12 16653 phicl2 16805 dec2dvds 17101 dec5dvds2 17103 modxai 17106 mod2xnegi 17109 gcdi 17111 gcdmodi 17112 1259lem1 17168 1259lem2 17169 1259lem3 17170 1259lem4 17171 1259lem5 17172 2503lem1 17174 2503lem2 17175 2503lem3 17176 4001lem1 17178 4001lem2 17179 4001lem3 17180 4001lem4 17181 ppi1i 27211 ppi2i 27212 ppiublem1 27246 konigsberglem5 30275 dp2lt10 32866 dp2ltc 32869 ballotlemfelz 34493 hgt750lemd 34663 hgt750lem 34666 hgt750leme 34673 poimirlem26 37653 poimirlem28 37655 fmtno4prmfac 47559 31prm 47584 nfermltl2rev 47730 linevalexample 48312 ackval42 48617 |
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