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Mirrors > Home > MPE Home > Th. List > qnnen | Structured version Visualization version GIF version |
Description: The rational numbers are countable. This proof does not use the Axiom of Choice, even though it uses an onto function, because the base set (ℤ × ℕ) is numerable. Exercise 2 of [Enderton] p. 133. For purposes of the Metamath 100 list, we are considering Mario Carneiro's revision as the date this proof was completed. This is Metamath 100 proof #3. (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 3-Mar-2013.) |
Ref | Expression |
---|---|
qnnen | ⊢ ℚ ≈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omelon 8707 | . . . . . . 7 ⊢ ω ∈ On | |
2 | nnenom 12987 | . . . . . . . 8 ⊢ ℕ ≈ ω | |
3 | 2 | ensymi 8159 | . . . . . . 7 ⊢ ω ≈ ℕ |
4 | isnumi 8972 | . . . . . . 7 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card) | |
5 | 1, 3, 4 | mp2an 672 | . . . . . 6 ⊢ ℕ ∈ dom card |
6 | znnen 15147 | . . . . . . 7 ⊢ ℤ ≈ ℕ | |
7 | ennum 8973 | . . . . . . 7 ⊢ (ℤ ≈ ℕ → (ℤ ∈ dom card ↔ ℕ ∈ dom card)) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ (ℤ ∈ dom card ↔ ℕ ∈ dom card) |
9 | 5, 8 | mpbir 221 | . . . . 5 ⊢ ℤ ∈ dom card |
10 | xpnum 8977 | . . . . 5 ⊢ ((ℤ ∈ dom card ∧ ℕ ∈ dom card) → (ℤ × ℕ) ∈ dom card) | |
11 | 9, 5, 10 | mp2an 672 | . . . 4 ⊢ (ℤ × ℕ) ∈ dom card |
12 | eqid 2771 | . . . . . 6 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) | |
13 | ovex 6823 | . . . . . 6 ⊢ (𝑥 / 𝑦) ∈ V | |
14 | 12, 13 | fnmpt2i 7389 | . . . . 5 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) Fn (ℤ × ℕ) |
15 | 12 | rnmpt2 6917 | . . . . . 6 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)} |
16 | elq 11993 | . . . . . . 7 ⊢ (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)) | |
17 | 16 | abbi2i 2887 | . . . . . 6 ⊢ ℚ = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)} |
18 | 15, 17 | eqtr4i 2796 | . . . . 5 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = ℚ |
19 | df-fo 6037 | . . . . 5 ⊢ ((𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)):(ℤ × ℕ)–onto→ℚ ↔ ((𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) Fn (ℤ × ℕ) ∧ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = ℚ)) | |
20 | 14, 18, 19 | mpbir2an 690 | . . . 4 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)):(ℤ × ℕ)–onto→ℚ |
21 | fodomnum 9080 | . . . 4 ⊢ ((ℤ × ℕ) ∈ dom card → ((𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)):(ℤ × ℕ)–onto→ℚ → ℚ ≼ (ℤ × ℕ))) | |
22 | 11, 20, 21 | mp2 9 | . . 3 ⊢ ℚ ≼ (ℤ × ℕ) |
23 | nnex 11228 | . . . . . 6 ⊢ ℕ ∈ V | |
24 | 23 | enref 8142 | . . . . 5 ⊢ ℕ ≈ ℕ |
25 | xpen 8279 | . . . . 5 ⊢ ((ℤ ≈ ℕ ∧ ℕ ≈ ℕ) → (ℤ × ℕ) ≈ (ℕ × ℕ)) | |
26 | 6, 24, 25 | mp2an 672 | . . . 4 ⊢ (ℤ × ℕ) ≈ (ℕ × ℕ) |
27 | xpnnen 15145 | . . . 4 ⊢ (ℕ × ℕ) ≈ ℕ | |
28 | 26, 27 | entri 8163 | . . 3 ⊢ (ℤ × ℕ) ≈ ℕ |
29 | domentr 8168 | . . 3 ⊢ ((ℚ ≼ (ℤ × ℕ) ∧ (ℤ × ℕ) ≈ ℕ) → ℚ ≼ ℕ) | |
30 | 22, 28, 29 | mp2an 672 | . 2 ⊢ ℚ ≼ ℕ |
31 | qex 12003 | . . 3 ⊢ ℚ ∈ V | |
32 | nnssq 12000 | . . 3 ⊢ ℕ ⊆ ℚ | |
33 | ssdomg 8155 | . . 3 ⊢ (ℚ ∈ V → (ℕ ⊆ ℚ → ℕ ≼ ℚ)) | |
34 | 31, 32, 33 | mp2 9 | . 2 ⊢ ℕ ≼ ℚ |
35 | sbth 8236 | . 2 ⊢ ((ℚ ≼ ℕ ∧ ℕ ≼ ℚ) → ℚ ≈ ℕ) | |
36 | 30, 34, 35 | mp2an 672 | 1 ⊢ ℚ ≈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1631 ∈ wcel 2145 {cab 2757 ∃wrex 3062 Vcvv 3351 ⊆ wss 3723 class class class wbr 4786 × cxp 5247 dom cdm 5249 ran crn 5250 Oncon0 5866 Fn wfn 6026 –onto→wfo 6029 (class class class)co 6793 ↦ cmpt2 6795 ωcom 7212 ≈ cen 8106 ≼ cdom 8107 cardccrd 8961 / cdiv 10886 ℕcn 11222 ℤcz 11579 ℚcq 11991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-omul 7718 df-er 7896 df-map 8011 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-oi 8571 df-card 8965 df-acn 8968 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-n0 11495 df-z 11580 df-uz 11889 df-q 11992 |
This theorem is referenced by: rpnnen 15162 resdomq 15179 re2ndc 22824 ovolq 23479 opnmblALT 23591 vitali 23601 mbfimaopnlem 23642 mbfaddlem 23647 mblfinlem1 33779 irrapx1 37918 qenom 40093 |
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