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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 16003 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 12382 | . . . 4 ⊢ ℕ ⊆ ℤ | |
2 | 0z 12372 | . . . . 5 ⊢ 0 ∈ ℤ | |
3 | 0nnn 12051 | . . . . 5 ⊢ ¬ 0 ∈ ℕ | |
4 | 2, 3 | pm3.2i 472 | . . . 4 ⊢ (0 ∈ ℤ ∧ ¬ 0 ∈ ℕ) |
5 | ssnelpss 4052 | . . . 4 ⊢ (ℕ ⊆ ℤ → ((0 ∈ ℤ ∧ ¬ 0 ∈ ℕ) → ℕ ⊊ ℤ)) | |
6 | 1, 4, 5 | mp2 9 | . . 3 ⊢ ℕ ⊊ ℤ |
7 | zssq 12738 | . . . 4 ⊢ ℤ ⊆ ℚ | |
8 | 1z 12392 | . . . . . 6 ⊢ 1 ∈ ℤ | |
9 | 2nn 12088 | . . . . . 6 ⊢ 2 ∈ ℕ | |
10 | znq 12734 | . . . . . 6 ⊢ ((1 ∈ ℤ ∧ 2 ∈ ℕ) → (1 / 2) ∈ ℚ) | |
11 | 8, 9, 10 | mp2an 690 | . . . . 5 ⊢ (1 / 2) ∈ ℚ |
12 | halfnz 12440 | . . . . 5 ⊢ ¬ (1 / 2) ∈ ℤ | |
13 | 11, 12 | pm3.2i 472 | . . . 4 ⊢ ((1 / 2) ∈ ℚ ∧ ¬ (1 / 2) ∈ ℤ) |
14 | ssnelpss 4052 | . . . 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 472 | . 2 ⊢ (ℕ ⊊ ℤ ∧ ℤ ⊊ ℚ) |
17 | qssre 12741 | . . . 4 ⊢ ℚ ⊆ ℝ | |
18 | sqrt2re 16000 | . . . . 5 ⊢ (√‘2) ∈ ℝ | |
19 | sqrt2irr 15999 | . . . . . 6 ⊢ (√‘2) ∉ ℚ | |
20 | 19 | neli 3049 | . . . . 5 ⊢ ¬ (√‘2) ∈ ℚ |
21 | 18, 20 | pm3.2i 472 | . . . 4 ⊢ ((√‘2) ∈ ℝ ∧ ¬ (√‘2) ∈ ℚ) |
22 | ssnelpss 4052 | . . . 4 ⊢ (ℚ ⊆ ℝ → (((√‘2) ∈ ℝ ∧ ¬ (√‘2) ∈ ℚ) → ℚ ⊊ ℝ)) | |
23 | 17, 21, 22 | mp2 9 | . . 3 ⊢ ℚ ⊊ ℝ |
24 | ax-resscn 10970 | . . . 4 ⊢ ℝ ⊆ ℂ | |
25 | ax-icn 10972 | . . . . 5 ⊢ i ∈ ℂ | |
26 | inelr 12005 | . . . . 5 ⊢ ¬ i ∈ ℝ | |
27 | 25, 26 | pm3.2i 472 | . . . 4 ⊢ (i ∈ ℂ ∧ ¬ i ∈ ℝ) |
28 | ssnelpss 4052 | . . . 4 ⊢ (ℝ ⊆ ℂ → ((i ∈ ℂ ∧ ¬ i ∈ ℝ) → ℝ ⊊ ℂ)) | |
29 | 24, 27, 28 | mp2 9 | . . 3 ⊢ ℝ ⊊ ℂ |
30 | 23, 29 | pm3.2i 472 | . 2 ⊢ (ℚ ⊊ ℝ ∧ ℝ ⊊ ℂ) |
31 | 16, 30 | pm3.2i 472 | 1 ⊢ ((ℕ ⊊ ℤ ∧ ℤ ⊊ ℚ) ∧ (ℚ ⊊ ℝ ∧ ℝ ⊊ ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 ∈ wcel 2104 ⊆ wss 3892 ⊊ wpss 3893 ‘cfv 6454 (class class class)co 7303 ℂcc 10911 ℝcr 10912 0cc0 10913 1c1 10914 ici 10915 / cdiv 11674 ℕcn 12015 2c2 12070 ℤcz 12361 ℚcq 12730 √csqrt 14985 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 ax-cnex 10969 ax-resscn 10970 ax-1cn 10971 ax-icn 10972 ax-addcl 10973 ax-addrcl 10974 ax-mulcl 10975 ax-mulrcl 10976 ax-mulcom 10977 ax-addass 10978 ax-mulass 10979 ax-distr 10980 ax-i2m1 10981 ax-1ne0 10982 ax-1rid 10983 ax-rnegex 10984 ax-rrecex 10985 ax-cnre 10986 ax-pre-lttri 10987 ax-pre-lttrn 10988 ax-pre-ltadd 10989 ax-pre-mulgt0 10990 ax-pre-sup 10991 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5496 df-eprel 5502 df-po 5510 df-so 5511 df-fr 5551 df-we 5553 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-pred 6213 df-ord 6280 df-on 6281 df-lim 6282 df-suc 6283 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-riota 7260 df-ov 7306 df-oprab 7307 df-mpo 7308 df-om 7741 df-1st 7859 df-2nd 7860 df-frecs 8124 df-wrecs 8155 df-recs 8229 df-rdg 8268 df-er 8525 df-en 8761 df-dom 8762 df-sdom 8763 df-sup 9241 df-pnf 11053 df-mnf 11054 df-xr 11055 df-ltxr 11056 df-le 11057 df-sub 11249 df-neg 11250 df-div 11675 df-nn 12016 df-2 12078 df-3 12079 df-n0 12276 df-z 12362 df-uz 12625 df-q 12731 df-rp 12773 df-seq 13764 df-exp 13825 df-cj 14851 df-re 14852 df-im 14853 df-sqrt 14987 df-abs 14988 |
This theorem is referenced by: (None) |
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