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Mirrors > Home > MPE Home > Th. List > nn0ssz | Structured version Visualization version GIF version |
Description: Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
nn0ssz | ⊢ ℕ0 ⊆ ℤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-n0 11485 | . 2 ⊢ ℕ0 = (ℕ ∪ {0}) | |
2 | nnssz 11589 | . . 3 ⊢ ℕ ⊆ ℤ | |
3 | 0z 11580 | . . . 4 ⊢ 0 ∈ ℤ | |
4 | c0ex 10226 | . . . . 5 ⊢ 0 ∈ V | |
5 | 4 | snss 4460 | . . . 4 ⊢ (0 ∈ ℤ ↔ {0} ⊆ ℤ) |
6 | 3, 5 | mpbi 220 | . . 3 ⊢ {0} ⊆ ℤ |
7 | 2, 6 | unssi 3931 | . 2 ⊢ (ℕ ∪ {0}) ⊆ ℤ |
8 | 1, 7 | eqsstri 3776 | 1 ⊢ ℕ0 ⊆ ℤ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2139 ∪ cun 3713 ⊆ wss 3715 {csn 4321 0cc0 10128 ℕcn 11212 ℕ0cn0 11484 ℤcz 11569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-n0 11485 df-z 11570 |
This theorem is referenced by: nn0z 11592 nn0zi 11594 nn0zd 11672 nn0ssq 11989 nthruz 15181 oddnn02np1 15274 evennn02n 15276 bitsf1ocnv 15368 eucalg 15502 pclem 15745 0ram 15926 0ram2 15927 0ramcl 15929 gexex 18456 iscmet3lem3 23288 plyeq0lem 24165 dgrlem 24184 archirngz 30052 diophrw 37824 diophin 37838 diophun 37839 eq0rabdioph 37842 eqrabdioph 37843 rabdiophlem1 37867 diophren 37879 etransclem48 41002 |
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