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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 16289 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 12635 | . . . 4 ⊢ ℕ ⊆ ℤ | |
| 2 | 0z 12624 | . . . . 5 ⊢ 0 ∈ ℤ | |
| 3 | 0nnn 12302 | . . . . 5 ⊢ ¬ 0 ∈ ℕ | |
| 4 | 2, 3 | pm3.2i 470 | . . . 4 ⊢ (0 ∈ ℤ ∧ ¬ 0 ∈ ℕ) |
| 5 | ssnelpss 4114 | . . . 4 ⊢ (ℕ ⊆ ℤ → ((0 ∈ ℤ ∧ ¬ 0 ∈ ℕ) → ℕ ⊊ ℤ)) | |
| 6 | 1, 4, 5 | mp2 9 | . . 3 ⊢ ℕ ⊊ ℤ |
| 7 | zssq 12998 | . . . 4 ⊢ ℤ ⊆ ℚ | |
| 8 | 1z 12647 | . . . . . 6 ⊢ 1 ∈ ℤ | |
| 9 | 2nn 12339 | . . . . . 6 ⊢ 2 ∈ ℕ | |
| 10 | znq 12994 | . . . . . 6 ⊢ ((1 ∈ ℤ ∧ 2 ∈ ℕ) → (1 / 2) ∈ ℚ) | |
| 11 | 8, 9, 10 | mp2an 692 | . . . . 5 ⊢ (1 / 2) ∈ ℚ |
| 12 | halfnz 12696 | . . . . 5 ⊢ ¬ (1 / 2) ∈ ℤ | |
| 13 | 11, 12 | pm3.2i 470 | . . . 4 ⊢ ((1 / 2) ∈ ℚ ∧ ¬ (1 / 2) ∈ ℤ) |
| 14 | ssnelpss 4114 | . . . 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 13001 | . . . 4 ⊢ ℚ ⊆ ℝ | |
| 18 | sqrt2re 16286 | . . . . 5 ⊢ (√‘2) ∈ ℝ | |
| 19 | sqrt2irr 16285 | . . . . . 6 ⊢ (√‘2) ∉ ℚ | |
| 20 | 19 | neli 3048 | . . . . 5 ⊢ ¬ (√‘2) ∈ ℚ |
| 21 | 18, 20 | pm3.2i 470 | . . . 4 ⊢ ((√‘2) ∈ ℝ ∧ ¬ (√‘2) ∈ ℚ) |
| 22 | ssnelpss 4114 | . . . 4 ⊢ (ℚ ⊆ ℝ → (((√‘2) ∈ ℝ ∧ ¬ (√‘2) ∈ ℚ) → ℚ ⊊ ℝ)) | |
| 23 | 17, 21, 22 | mp2 9 | . . 3 ⊢ ℚ ⊊ ℝ |
| 24 | ax-resscn 11212 | . . . 4 ⊢ ℝ ⊆ ℂ | |
| 25 | ax-icn 11214 | . . . . 5 ⊢ i ∈ ℂ | |
| 26 | inelr 12256 | . . . . 5 ⊢ ¬ i ∈ ℝ | |
| 27 | 25, 26 | pm3.2i 470 | . . . 4 ⊢ (i ∈ ℂ ∧ ¬ i ∈ ℝ) |
| 28 | ssnelpss 4114 | . . . 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 395 ∈ wcel 2108 ⊆ wss 3951 ⊊ wpss 3952 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 ici 11157 / cdiv 11920 ℕcn 12266 2c2 12321 ℤcz 12613 ℚcq 12990 √csqrt 15272 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 |
| This theorem is referenced by: (None) |
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