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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 16285. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
nthruz | ⊢ (ℕ ⊊ ℕ0 ∧ ℕ0 ⊊ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnssnn0 12527 | . . 3 ⊢ ℕ ⊆ ℕ0 | |
2 | 0nn0 12539 | . . . 4 ⊢ 0 ∈ ℕ0 | |
3 | 0nnn 12300 | . . . 4 ⊢ ¬ 0 ∈ ℕ | |
4 | 2, 3 | pm3.2i 470 | . . 3 ⊢ (0 ∈ ℕ0 ∧ ¬ 0 ∈ ℕ) |
5 | ssnelpss 4124 | . . 3 ⊢ (ℕ ⊆ ℕ0 → ((0 ∈ ℕ0 ∧ ¬ 0 ∈ ℕ) → ℕ ⊊ ℕ0)) | |
6 | 1, 4, 5 | mp2 9 | . 2 ⊢ ℕ ⊊ ℕ0 |
7 | nn0ssz 12634 | . . 3 ⊢ ℕ0 ⊆ ℤ | |
8 | neg1z 12651 | . . . 4 ⊢ -1 ∈ ℤ | |
9 | neg1lt0 12381 | . . . . 5 ⊢ -1 < 0 | |
10 | nn0nlt0 12550 | . . . . 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 4124 | . . 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 2106 ⊆ wss 3963 ⊊ wpss 3964 class class class wbr 5148 0cc0 11153 1c1 11154 < clt 11293 -cneg 11491 ℕcn 12264 ℕ0cn0 12524 ℤcz 12611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 |
This theorem is referenced by: (None) |
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