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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 16141. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
nthruz | ⊢ (ℕ ⊊ ℕ0 ∧ ℕ0 ⊊ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnssnn0 12423 | . . 3 ⊢ ℕ ⊆ ℕ0 | |
2 | 0nn0 12435 | . . . 4 ⊢ 0 ∈ ℕ0 | |
3 | 0nnn 12196 | . . . 4 ⊢ ¬ 0 ∈ ℕ | |
4 | 2, 3 | pm3.2i 472 | . . 3 ⊢ (0 ∈ ℕ0 ∧ ¬ 0 ∈ ℕ) |
5 | ssnelpss 4076 | . . 3 ⊢ (ℕ ⊆ ℕ0 → ((0 ∈ ℕ0 ∧ ¬ 0 ∈ ℕ) → ℕ ⊊ ℕ0)) | |
6 | 1, 4, 5 | mp2 9 | . 2 ⊢ ℕ ⊊ ℕ0 |
7 | nn0ssz 12529 | . . 3 ⊢ ℕ0 ⊆ ℤ | |
8 | neg1z 12546 | . . . 4 ⊢ -1 ∈ ℤ | |
9 | neg1lt0 12277 | . . . . 5 ⊢ -1 < 0 | |
10 | nn0nlt0 12446 | . . . . 5 ⊢ (-1 ∈ ℕ0 → ¬ -1 < 0) | |
11 | 9, 10 | mt2 199 | . . . 4 ⊢ ¬ -1 ∈ ℕ0 |
12 | 8, 11 | pm3.2i 472 | . . 3 ⊢ (-1 ∈ ℤ ∧ ¬ -1 ∈ ℕ0) |
13 | ssnelpss 4076 | . . 3 ⊢ (ℕ0 ⊆ ℤ → ((-1 ∈ ℤ ∧ ¬ -1 ∈ ℕ0) → ℕ0 ⊊ ℤ)) | |
14 | 7, 12, 13 | mp2 9 | . 2 ⊢ ℕ0 ⊊ ℤ |
15 | 6, 14 | pm3.2i 472 | 1 ⊢ (ℕ ⊊ ℕ0 ∧ ℕ0 ⊊ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 ∈ wcel 2107 ⊆ wss 3915 ⊊ wpss 3916 class class class wbr 5110 0cc0 11058 1c1 11059 < clt 11196 -cneg 11393 ℕcn 12160 ℕ0cn0 12420 ℤcz 12506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-n0 12421 df-z 12507 |
This theorem is referenced by: (None) |
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