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Mirrors > Home > ILE Home > Th. List > nn0ssz | 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 9212 | . 2 ⊢ ℕ0 = (ℕ ∪ {0}) | |
2 | nnssz 9305 | . . 3 ⊢ ℕ ⊆ ℤ | |
3 | 0z 9299 | . . . 4 ⊢ 0 ∈ ℤ | |
4 | c0ex 7986 | . . . . 5 ⊢ 0 ∈ V | |
5 | 4 | snss 3745 | . . . 4 ⊢ (0 ∈ ℤ ↔ {0} ⊆ ℤ) |
6 | 3, 5 | mpbi 145 | . . 3 ⊢ {0} ⊆ ℤ |
7 | 2, 6 | unssi 3325 | . 2 ⊢ (ℕ ∪ {0}) ⊆ ℤ |
8 | 1, 7 | eqsstri 3202 | 1 ⊢ ℕ0 ⊆ ℤ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 ∪ cun 3142 ⊆ wss 3144 {csn 3610 0cc0 7846 ℕcn 8954 ℕ0cn0 9211 ℤcz 9288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-cnex 7937 ax-resscn 7938 ax-1cn 7939 ax-1re 7940 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-addcom 7946 ax-addass 7948 ax-distr 7950 ax-i2m1 7951 ax-0lt1 7952 ax-0id 7954 ax-rnegex 7955 ax-cnre 7957 ax-pre-ltirr 7958 ax-pre-ltwlin 7959 ax-pre-lttrn 7960 ax-pre-ltadd 7962 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-br 4022 df-opab 4083 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-iota 5199 df-fun 5240 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-pnf 8029 df-mnf 8030 df-xr 8031 df-ltxr 8032 df-le 8033 df-sub 8165 df-neg 8166 df-inn 8955 df-n0 9212 df-z 9289 |
This theorem is referenced by: nn0z 9308 nn0zi 9310 nn0zd 9408 nn0ssq 9664 oddnn02np1 11926 evennn02n 11928 eulerthlemrprm 12272 eulerthlema 12273 eulerthlemh 12274 eulerthlemth 12275 pcprecl 12332 pcprendvds 12333 pcpremul 12336 |
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