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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 16286 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 12633 | . . . 4 ⊢ ℕ ⊆ ℤ | |
2 | 0z 12622 | . . . . 5 ⊢ 0 ∈ ℤ | |
3 | 0nnn 12300 | . . . . 5 ⊢ ¬ 0 ∈ ℕ | |
4 | 2, 3 | pm3.2i 470 | . . . 4 ⊢ (0 ∈ ℤ ∧ ¬ 0 ∈ ℕ) |
5 | ssnelpss 4124 | . . . 4 ⊢ (ℕ ⊆ ℤ → ((0 ∈ ℤ ∧ ¬ 0 ∈ ℕ) → ℕ ⊊ ℤ)) | |
6 | 1, 4, 5 | mp2 9 | . . 3 ⊢ ℕ ⊊ ℤ |
7 | zssq 12996 | . . . 4 ⊢ ℤ ⊆ ℚ | |
8 | 1z 12645 | . . . . . 6 ⊢ 1 ∈ ℤ | |
9 | 2nn 12337 | . . . . . 6 ⊢ 2 ∈ ℕ | |
10 | znq 12992 | . . . . . 6 ⊢ ((1 ∈ ℤ ∧ 2 ∈ ℕ) → (1 / 2) ∈ ℚ) | |
11 | 8, 9, 10 | mp2an 692 | . . . . 5 ⊢ (1 / 2) ∈ ℚ |
12 | halfnz 12694 | . . . . 5 ⊢ ¬ (1 / 2) ∈ ℤ | |
13 | 11, 12 | pm3.2i 470 | . . . 4 ⊢ ((1 / 2) ∈ ℚ ∧ ¬ (1 / 2) ∈ ℤ) |
14 | ssnelpss 4124 | . . . 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 12999 | . . . 4 ⊢ ℚ ⊆ ℝ | |
18 | sqrt2re 16283 | . . . . 5 ⊢ (√‘2) ∈ ℝ | |
19 | sqrt2irr 16282 | . . . . . 6 ⊢ (√‘2) ∉ ℚ | |
20 | 19 | neli 3046 | . . . . 5 ⊢ ¬ (√‘2) ∈ ℚ |
21 | 18, 20 | pm3.2i 470 | . . . 4 ⊢ ((√‘2) ∈ ℝ ∧ ¬ (√‘2) ∈ ℚ) |
22 | ssnelpss 4124 | . . . 4 ⊢ (ℚ ⊆ ℝ → (((√‘2) ∈ ℝ ∧ ¬ (√‘2) ∈ ℚ) → ℚ ⊊ ℝ)) | |
23 | 17, 21, 22 | mp2 9 | . . 3 ⊢ ℚ ⊊ ℝ |
24 | ax-resscn 11210 | . . . 4 ⊢ ℝ ⊆ ℂ | |
25 | ax-icn 11212 | . . . . 5 ⊢ i ∈ ℂ | |
26 | inelr 12254 | . . . . 5 ⊢ ¬ i ∈ ℝ | |
27 | 25, 26 | pm3.2i 470 | . . . 4 ⊢ (i ∈ ℂ ∧ ¬ i ∈ ℝ) |
28 | ssnelpss 4124 | . . . 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 2106 ⊆ wss 3963 ⊊ wpss 3964 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 ici 11155 / cdiv 11918 ℕcn 12264 2c2 12319 ℤcz 12611 ℚcq 12988 √csqrt 15269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 |
This theorem is referenced by: (None) |
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