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Mirrors > Home > MPE Home > Th. List > nthruc | Structured version Visualization version GIF version |
Description: The sequence ℕ, ℤ, ℚ, ℝ, and ℂ 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 ℤ but not ℕ, one-half belongs to ℚ but not ℤ, the square root of 2 belongs to ℝ but not ℚ, and finally that the imaginary number i belongs to ℂ but not ℝ. See nthruz 15594 for a further refinement. (Contributed by NM, 12-Jan-2002.) |
Ref | Expression |
---|---|
nthruc | ⊢ ((ℕ ⊊ ℤ ∧ ℤ ⊊ ℚ) ∧ (ℚ ⊊ ℝ ∧ ℝ ⊊ ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnssz 11990 | . . . 4 ⊢ ℕ ⊆ ℤ | |
2 | 0z 11980 | . . . . 5 ⊢ 0 ∈ ℤ | |
3 | 0nnn 11661 | . . . . 5 ⊢ ¬ 0 ∈ ℕ | |
4 | 2, 3 | pm3.2i 471 | . . . 4 ⊢ (0 ∈ ℤ ∧ ¬ 0 ∈ ℕ) |
5 | ssnelpss 4085 | . . . 4 ⊢ (ℕ ⊆ ℤ → ((0 ∈ ℤ ∧ ¬ 0 ∈ ℕ) → ℕ ⊊ ℤ)) | |
6 | 1, 4, 5 | mp2 9 | . . 3 ⊢ ℕ ⊊ ℤ |
7 | zssq 12343 | . . . 4 ⊢ ℤ ⊆ ℚ | |
8 | 1z 12000 | . . . . . 6 ⊢ 1 ∈ ℤ | |
9 | 2nn 11698 | . . . . . 6 ⊢ 2 ∈ ℕ | |
10 | znq 12340 | . . . . . 6 ⊢ ((1 ∈ ℤ ∧ 2 ∈ ℕ) → (1 / 2) ∈ ℚ) | |
11 | 8, 9, 10 | mp2an 688 | . . . . 5 ⊢ (1 / 2) ∈ ℚ |
12 | halfnz 12048 | . . . . 5 ⊢ ¬ (1 / 2) ∈ ℤ | |
13 | 11, 12 | pm3.2i 471 | . . . 4 ⊢ ((1 / 2) ∈ ℚ ∧ ¬ (1 / 2) ∈ ℤ) |
14 | ssnelpss 4085 | . . . 4 ⊢ (ℤ ⊆ ℚ → (((1 / 2) ∈ ℚ ∧ ¬ (1 / 2) ∈ ℤ) → ℤ ⊊ ℚ)) | |
15 | 7, 13, 14 | mp2 9 | . . 3 ⊢ ℤ ⊊ ℚ |
16 | 6, 15 | pm3.2i 471 | . 2 ⊢ (ℕ ⊊ ℤ ∧ ℤ ⊊ ℚ) |
17 | qssre 12346 | . . . 4 ⊢ ℚ ⊆ ℝ | |
18 | sqrt2re 15591 | . . . . 5 ⊢ (√‘2) ∈ ℝ | |
19 | sqrt2irr 15590 | . . . . . 6 ⊢ (√‘2) ∉ ℚ | |
20 | 19 | neli 3122 | . . . . 5 ⊢ ¬ (√‘2) ∈ ℚ |
21 | 18, 20 | pm3.2i 471 | . . . 4 ⊢ ((√‘2) ∈ ℝ ∧ ¬ (√‘2) ∈ ℚ) |
22 | ssnelpss 4085 | . . . 4 ⊢ (ℚ ⊆ ℝ → (((√‘2) ∈ ℝ ∧ ¬ (√‘2) ∈ ℚ) → ℚ ⊊ ℝ)) | |
23 | 17, 21, 22 | mp2 9 | . . 3 ⊢ ℚ ⊊ ℝ |
24 | ax-resscn 10582 | . . . 4 ⊢ ℝ ⊆ ℂ | |
25 | ax-icn 10584 | . . . . 5 ⊢ i ∈ ℂ | |
26 | inelr 11616 | . . . . 5 ⊢ ¬ i ∈ ℝ | |
27 | 25, 26 | pm3.2i 471 | . . . 4 ⊢ (i ∈ ℂ ∧ ¬ i ∈ ℝ) |
28 | ssnelpss 4085 | . . . 4 ⊢ (ℝ ⊆ ℂ → ((i ∈ ℂ ∧ ¬ i ∈ ℝ) → ℝ ⊊ ℂ)) | |
29 | 24, 27, 28 | mp2 9 | . . 3 ⊢ ℝ ⊊ ℂ |
30 | 23, 29 | pm3.2i 471 | . 2 ⊢ (ℚ ⊊ ℝ ∧ ℝ ⊊ ℂ) |
31 | 16, 30 | pm3.2i 471 | 1 ⊢ ((ℕ ⊊ ℤ ∧ ℤ ⊊ ℚ) ∧ (ℚ ⊊ ℝ ∧ ℝ ⊊ ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 ∈ wcel 2105 ⊆ wss 3933 ⊊ wpss 3934 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 ℝcr 10524 0cc0 10525 1c1 10526 ici 10527 / cdiv 11285 ℕcn 11626 2c2 11680 ℤcz 11969 ℚcq 12336 √csqrt 14580 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 |
This theorem is referenced by: (None) |
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