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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 14982 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 11397 | . . . 4 ⊢ ℕ ⊆ ℤ | |
2 | 0z 11388 | . . . . 5 ⊢ 0 ∈ ℤ | |
3 | 0nnn 11052 | . . . . 5 ⊢ ¬ 0 ∈ ℕ | |
4 | 2, 3 | pm3.2i 471 | . . . 4 ⊢ (0 ∈ ℤ ∧ ¬ 0 ∈ ℕ) |
5 | ssnelpss 3718 | . . . 4 ⊢ (ℕ ⊆ ℤ → ((0 ∈ ℤ ∧ ¬ 0 ∈ ℕ) → ℕ ⊊ ℤ)) | |
6 | 1, 4, 5 | mp2 9 | . . 3 ⊢ ℕ ⊊ ℤ |
7 | zssq 11795 | . . . 4 ⊢ ℤ ⊆ ℚ | |
8 | 1z 11407 | . . . . . 6 ⊢ 1 ∈ ℤ | |
9 | 2nn 11185 | . . . . . 6 ⊢ 2 ∈ ℕ | |
10 | znq 11792 | . . . . . 6 ⊢ ((1 ∈ ℤ ∧ 2 ∈ ℕ) → (1 / 2) ∈ ℚ) | |
11 | 8, 9, 10 | mp2an 708 | . . . . 5 ⊢ (1 / 2) ∈ ℚ |
12 | halfnz 11455 | . . . . 5 ⊢ ¬ (1 / 2) ∈ ℤ | |
13 | 11, 12 | pm3.2i 471 | . . . 4 ⊢ ((1 / 2) ∈ ℚ ∧ ¬ (1 / 2) ∈ ℤ) |
14 | ssnelpss 3718 | . . . 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 11798 | . . . 4 ⊢ ℚ ⊆ ℝ | |
18 | sqrt2re 14980 | . . . . 5 ⊢ (√‘2) ∈ ℝ | |
19 | sqrt2irr 14979 | . . . . . 6 ⊢ (√‘2) ∉ ℚ | |
20 | 19 | neli 2899 | . . . . 5 ⊢ ¬ (√‘2) ∈ ℚ |
21 | 18, 20 | pm3.2i 471 | . . . 4 ⊢ ((√‘2) ∈ ℝ ∧ ¬ (√‘2) ∈ ℚ) |
22 | ssnelpss 3718 | . . . 4 ⊢ (ℚ ⊆ ℝ → (((√‘2) ∈ ℝ ∧ ¬ (√‘2) ∈ ℚ) → ℚ ⊊ ℝ)) | |
23 | 17, 21, 22 | mp2 9 | . . 3 ⊢ ℚ ⊊ ℝ |
24 | ax-resscn 9993 | . . . 4 ⊢ ℝ ⊆ ℂ | |
25 | ax-icn 9995 | . . . . 5 ⊢ i ∈ ℂ | |
26 | inelr 11010 | . . . . 5 ⊢ ¬ i ∈ ℝ | |
27 | 25, 26 | pm3.2i 471 | . . . 4 ⊢ (i ∈ ℂ ∧ ¬ i ∈ ℝ) |
28 | ssnelpss 3718 | . . . 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 384 ∈ wcel 1990 ⊆ wss 3574 ⊊ wpss 3575 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 ici 9938 / cdiv 10684 ℕcn 11020 2c2 11070 ℤcz 11377 ℚcq 11788 √csqrt 13973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 |
This theorem is referenced by: (None) |
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