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| Description: The sequence ℕ, ℕ0, and ℤ forms a chain of proper subsets. In each case the proper subset relationship is shown by demonstrating a number that belongs to one set but not the other. We show that zero belongs to ℕ0 but not ℕ and minus one belongs to ℤ but not ℕ0. This theorem refines the chain of proper subsets nthruc 16289. (Contributed by NM, 9-May-2004.) | 
| Ref | Expression | 
|---|---|
| nthruz | ⊢ (ℕ ⊊ ℕ0 ∧ ℕ0 ⊊ ℤ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nnssnn0 12531 | . . 3 ⊢ ℕ ⊆ ℕ0 | |
| 2 | 0nn0 12543 | . . . 4 ⊢ 0 ∈ ℕ0 | |
| 3 | 0nnn 12303 | . . . 4 ⊢ ¬ 0 ∈ ℕ | |
| 4 | 2, 3 | pm3.2i 470 | . . 3 ⊢ (0 ∈ ℕ0 ∧ ¬ 0 ∈ ℕ) | 
| 5 | ssnelpss 4113 | . . 3 ⊢ (ℕ ⊆ ℕ0 → ((0 ∈ ℕ0 ∧ ¬ 0 ∈ ℕ) → ℕ ⊊ ℕ0)) | |
| 6 | 1, 4, 5 | mp2 9 | . 2 ⊢ ℕ ⊊ ℕ0 | 
| 7 | nn0ssz 12638 | . . 3 ⊢ ℕ0 ⊆ ℤ | |
| 8 | neg1z 12655 | . . . 4 ⊢ -1 ∈ ℤ | |
| 9 | neg1lt0 12384 | . . . . 5 ⊢ -1 < 0 | |
| 10 | nn0nlt0 12554 | . . . . 5 ⊢ (-1 ∈ ℕ0 → ¬ -1 < 0) | |
| 11 | 9, 10 | mt2 200 | . . . 4 ⊢ ¬ -1 ∈ ℕ0 | 
| 12 | 8, 11 | pm3.2i 470 | . . 3 ⊢ (-1 ∈ ℤ ∧ ¬ -1 ∈ ℕ0) | 
| 13 | ssnelpss 4113 | . . 3 ⊢ (ℕ0 ⊆ ℤ → ((-1 ∈ ℤ ∧ ¬ -1 ∈ ℕ0) → ℕ0 ⊊ ℤ)) | |
| 14 | 7, 12, 13 | mp2 9 | . 2 ⊢ ℕ0 ⊊ ℤ | 
| 15 | 6, 14 | pm3.2i 470 | 1 ⊢ (ℕ ⊊ ℕ0 ∧ ℕ0 ⊊ ℤ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∧ wa 395 ∈ wcel 2107 ⊆ wss 3950 ⊊ wpss 3951 class class class wbr 5142 0cc0 11156 1c1 11157 < clt 11296 -cneg 11494 ℕcn 12267 ℕ0cn0 12528 ℤcz 12615 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-n0 12529 df-z 12616 | 
| This theorem is referenced by: (None) | 
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