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Mirrors > Home > MPE Home > Th. List > nthruz | Structured version Visualization version GIF version |
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 16229. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
nthruz | ⊢ (ℕ ⊊ ℕ0 ∧ ℕ0 ⊊ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnssnn0 12506 | . . 3 ⊢ ℕ ⊆ ℕ0 | |
2 | 0nn0 12518 | . . . 4 ⊢ 0 ∈ ℕ0 | |
3 | 0nnn 12279 | . . . 4 ⊢ ¬ 0 ∈ ℕ | |
4 | 2, 3 | pm3.2i 470 | . . 3 ⊢ (0 ∈ ℕ0 ∧ ¬ 0 ∈ ℕ) |
5 | ssnelpss 4109 | . . 3 ⊢ (ℕ ⊆ ℕ0 → ((0 ∈ ℕ0 ∧ ¬ 0 ∈ ℕ) → ℕ ⊊ ℕ0)) | |
6 | 1, 4, 5 | mp2 9 | . 2 ⊢ ℕ ⊊ ℕ0 |
7 | nn0ssz 12612 | . . 3 ⊢ ℕ0 ⊆ ℤ | |
8 | neg1z 12629 | . . . 4 ⊢ -1 ∈ ℤ | |
9 | neg1lt0 12360 | . . . . 5 ⊢ -1 < 0 | |
10 | nn0nlt0 12529 | . . . . 5 ⊢ (-1 ∈ ℕ0 → ¬ -1 < 0) | |
11 | 9, 10 | mt2 199 | . . . 4 ⊢ ¬ -1 ∈ ℕ0 |
12 | 8, 11 | pm3.2i 470 | . . 3 ⊢ (-1 ∈ ℤ ∧ ¬ -1 ∈ ℕ0) |
13 | ssnelpss 4109 | . . 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 2099 ⊆ wss 3947 ⊊ wpss 3948 class class class wbr 5148 0cc0 11139 1c1 11140 < clt 11279 -cneg 11476 ℕcn 12243 ℕ0cn0 12503 ℤcz 12589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 |
This theorem is referenced by: (None) |
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