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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 9129 | . 2 ⊢ ℕ0 = (ℕ ∪ {0}) | |
2 | nnssz 9222 | . . 3 ⊢ ℕ ⊆ ℤ | |
3 | 0z 9216 | . . . 4 ⊢ 0 ∈ ℤ | |
4 | c0ex 7907 | . . . . 5 ⊢ 0 ∈ V | |
5 | 4 | snss 3707 | . . . 4 ⊢ (0 ∈ ℤ ↔ {0} ⊆ ℤ) |
6 | 3, 5 | mpbi 144 | . . 3 ⊢ {0} ⊆ ℤ |
7 | 2, 6 | unssi 3302 | . 2 ⊢ (ℕ ∪ {0}) ⊆ ℤ |
8 | 1, 7 | eqsstri 3179 | 1 ⊢ ℕ0 ⊆ ℤ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2141 ∪ cun 3119 ⊆ wss 3121 {csn 3581 0cc0 7767 ℕcn 8871 ℕ0cn0 9128 ℤcz 9205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-addcom 7867 ax-addass 7869 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-0id 7875 ax-rnegex 7876 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-ltadd 7883 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-inn 8872 df-n0 9129 df-z 9206 |
This theorem is referenced by: nn0z 9225 nn0zi 9227 nn0zd 9325 nn0ssq 9580 oddnn02np1 11832 evennn02n 11834 eulerthlemrprm 12176 eulerthlema 12177 eulerthlemh 12178 eulerthlemth 12179 pcprecl 12236 pcprendvds 12237 pcpremul 12240 |
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