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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 14820 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 11274 | . . . 4 ⊢ ℕ ⊆ ℤ | |
2 | 0z 11265 | . . . . 5 ⊢ 0 ∈ ℤ | |
3 | 0nnn 10929 | . . . . 5 ⊢ ¬ 0 ∈ ℕ | |
4 | 2, 3 | pm3.2i 470 | . . . 4 ⊢ (0 ∈ ℤ ∧ ¬ 0 ∈ ℕ) |
5 | ssnelpss 3680 | . . . 4 ⊢ (ℕ ⊆ ℤ → ((0 ∈ ℤ ∧ ¬ 0 ∈ ℕ) → ℕ ⊊ ℤ)) | |
6 | 1, 4, 5 | mp2 9 | . . 3 ⊢ ℕ ⊊ ℤ |
7 | zssq 11671 | . . . 4 ⊢ ℤ ⊆ ℚ | |
8 | 1z 11284 | . . . . . 6 ⊢ 1 ∈ ℤ | |
9 | 2nn 11062 | . . . . . 6 ⊢ 2 ∈ ℕ | |
10 | znq 11668 | . . . . . 6 ⊢ ((1 ∈ ℤ ∧ 2 ∈ ℕ) → (1 / 2) ∈ ℚ) | |
11 | 8, 9, 10 | mp2an 704 | . . . . 5 ⊢ (1 / 2) ∈ ℚ |
12 | halfnz 11331 | . . . . 5 ⊢ ¬ (1 / 2) ∈ ℤ | |
13 | 11, 12 | pm3.2i 470 | . . . 4 ⊢ ((1 / 2) ∈ ℚ ∧ ¬ (1 / 2) ∈ ℤ) |
14 | ssnelpss 3680 | . . . 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 470 | . 2 ⊢ (ℕ ⊊ ℤ ∧ ℤ ⊊ ℚ) |
17 | qssre 11674 | . . . 4 ⊢ ℚ ⊆ ℝ | |
18 | sqrt2re 14818 | . . . . 5 ⊢ (√‘2) ∈ ℝ | |
19 | sqrt2irr 14817 | . . . . . 6 ⊢ (√‘2) ∉ ℚ | |
20 | 19 | neli 2885 | . . . . 5 ⊢ ¬ (√‘2) ∈ ℚ |
21 | 18, 20 | pm3.2i 470 | . . . 4 ⊢ ((√‘2) ∈ ℝ ∧ ¬ (√‘2) ∈ ℚ) |
22 | ssnelpss 3680 | . . . 4 ⊢ (ℚ ⊆ ℝ → (((√‘2) ∈ ℝ ∧ ¬ (√‘2) ∈ ℚ) → ℚ ⊊ ℝ)) | |
23 | 17, 21, 22 | mp2 9 | . . 3 ⊢ ℚ ⊊ ℝ |
24 | ax-resscn 9872 | . . . 4 ⊢ ℝ ⊆ ℂ | |
25 | ax-icn 9874 | . . . . 5 ⊢ i ∈ ℂ | |
26 | inelr 10887 | . . . . 5 ⊢ ¬ i ∈ ℝ | |
27 | 25, 26 | pm3.2i 470 | . . . 4 ⊢ (i ∈ ℂ ∧ ¬ i ∈ ℝ) |
28 | ssnelpss 3680 | . . . 4 ⊢ (ℝ ⊆ ℂ → ((i ∈ ℂ ∧ ¬ i ∈ ℝ) → ℝ ⊊ ℂ)) | |
29 | 24, 27, 28 | mp2 9 | . . 3 ⊢ ℝ ⊊ ℂ |
30 | 23, 29 | pm3.2i 470 | . 2 ⊢ (ℚ ⊊ ℝ ∧ ℝ ⊊ ℂ) |
31 | 16, 30 | pm3.2i 470 | 1 ⊢ ((ℕ ⊊ ℤ ∧ ℤ ⊊ ℚ) ∧ (ℚ ⊊ ℝ ∧ ℝ ⊊ ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 ∈ wcel 1977 ⊆ wss 3540 ⊊ wpss 3541 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 ici 9817 / cdiv 10563 ℕcn 10897 2c2 10947 ℤcz 11254 ℚcq 11664 √csqrt 13821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 |
This theorem is referenced by: (None) |
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