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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 9637 | . . . . . . 7 ⊢ ω ∈ On | |
2 | nnenom 13941 | . . . . . . . 8 ⊢ ℕ ≈ ω | |
3 | 2 | ensymi 8996 | . . . . . . 7 ⊢ ω ≈ ℕ |
4 | isnumi 9937 | . . . . . . 7 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card) | |
5 | 1, 3, 4 | mp2an 690 | . . . . . 6 ⊢ ℕ ∈ dom card |
6 | znnen 16151 | . . . . . . 7 ⊢ ℤ ≈ ℕ | |
7 | ennum 9938 | . . . . . . 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 230 | . . . . 5 ⊢ ℤ ∈ dom card |
10 | xpnum 9942 | . . . . 5 ⊢ ((ℤ ∈ dom card ∧ ℕ ∈ dom card) → (ℤ × ℕ) ∈ dom card) | |
11 | 9, 5, 10 | mp2an 690 | . . . 4 ⊢ (ℤ × ℕ) ∈ dom card |
12 | eqid 2732 | . . . . . 6 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) | |
13 | ovex 7438 | . . . . . 6 ⊢ (𝑥 / 𝑦) ∈ V | |
14 | 12, 13 | fnmpoi 8052 | . . . . 5 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) Fn (ℤ × ℕ) |
15 | 12 | rnmpo 7538 | . . . . . 6 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)} |
16 | elq 12930 | . . . . . . 7 ⊢ (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)) | |
17 | 16 | eqabi 2869 | . . . . . 6 ⊢ ℚ = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)} |
18 | 15, 17 | eqtr4i 2763 | . . . . 5 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = ℚ |
19 | df-fo 6546 | . . . . 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 709 | . . . 4 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)):(ℤ × ℕ)–onto→ℚ |
21 | fodomnum 10048 | . . . 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 12214 | . . . . . 6 ⊢ ℕ ∈ V | |
24 | 23 | enref 8977 | . . . . 5 ⊢ ℕ ≈ ℕ |
25 | xpen 9136 | . . . . 5 ⊢ ((ℤ ≈ ℕ ∧ ℕ ≈ ℕ) → (ℤ × ℕ) ≈ (ℕ × ℕ)) | |
26 | 6, 24, 25 | mp2an 690 | . . . 4 ⊢ (ℤ × ℕ) ≈ (ℕ × ℕ) |
27 | xpnnen 16150 | . . . 4 ⊢ (ℕ × ℕ) ≈ ℕ | |
28 | 26, 27 | entri 9000 | . . 3 ⊢ (ℤ × ℕ) ≈ ℕ |
29 | domentr 9005 | . . 3 ⊢ ((ℚ ≼ (ℤ × ℕ) ∧ (ℤ × ℕ) ≈ ℕ) → ℚ ≼ ℕ) | |
30 | 22, 28, 29 | mp2an 690 | . 2 ⊢ ℚ ≼ ℕ |
31 | qex 12941 | . . 3 ⊢ ℚ ∈ V | |
32 | nnssq 12938 | . . 3 ⊢ ℕ ⊆ ℚ | |
33 | ssdomg 8992 | . . 3 ⊢ (ℚ ∈ V → (ℕ ⊆ ℚ → ℕ ≼ ℚ)) | |
34 | 31, 32, 33 | mp2 9 | . 2 ⊢ ℕ ≼ ℚ |
35 | sbth 9089 | . 2 ⊢ ((ℚ ≼ ℕ ∧ ℕ ≼ ℚ) → ℚ ≈ ℕ) | |
36 | 30, 34, 35 | mp2an 690 | 1 ⊢ ℚ ≈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∈ wcel 2106 {cab 2709 ∃wrex 3070 Vcvv 3474 ⊆ wss 3947 class class class wbr 5147 × cxp 5673 dom cdm 5675 ran crn 5676 Oncon0 6361 Fn wfn 6535 –onto→wfo 6538 (class class class)co 7405 ∈ cmpo 7407 ωcom 7851 ≈ cen 8932 ≼ cdom 8933 cardccrd 9926 / cdiv 11867 ℕcn 12208 ℤcz 12554 ℚcq 12928 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-omul 8467 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-oi 9501 df-card 9930 df-acn 9933 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 |
This theorem is referenced by: rpnnen 16166 resdomq 16183 re2ndc 24308 ovolq 24999 opnmblALT 25111 vitali 25121 mbfimaopnlem 25163 mbfaddlem 25168 mblfinlem1 36513 irrapx1 41551 qenom 44057 |
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