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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 9261 | . . . . . . 7 ⊢ ω ∈ On | |
2 | nnenom 13553 | . . . . . . . 8 ⊢ ℕ ≈ ω | |
3 | 2 | ensymi 8678 | . . . . . . 7 ⊢ ω ≈ ℕ |
4 | isnumi 9562 | . . . . . . 7 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card) | |
5 | 1, 3, 4 | mp2an 692 | . . . . . 6 ⊢ ℕ ∈ dom card |
6 | znnen 15773 | . . . . . . 7 ⊢ ℤ ≈ ℕ | |
7 | ennum 9563 | . . . . . . 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 234 | . . . . 5 ⊢ ℤ ∈ dom card |
10 | xpnum 9567 | . . . . 5 ⊢ ((ℤ ∈ dom card ∧ ℕ ∈ dom card) → (ℤ × ℕ) ∈ dom card) | |
11 | 9, 5, 10 | mp2an 692 | . . . 4 ⊢ (ℤ × ℕ) ∈ dom card |
12 | eqid 2737 | . . . . . 6 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) | |
13 | ovex 7246 | . . . . . 6 ⊢ (𝑥 / 𝑦) ∈ V | |
14 | 12, 13 | fnmpoi 7840 | . . . . 5 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) Fn (ℤ × ℕ) |
15 | 12 | rnmpo 7343 | . . . . . 6 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)} |
16 | elq 12546 | . . . . . . 7 ⊢ (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)) | |
17 | 16 | abbi2i 2876 | . . . . . 6 ⊢ ℚ = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)} |
18 | 15, 17 | eqtr4i 2768 | . . . . 5 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = ℚ |
19 | df-fo 6386 | . . . . 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 711 | . . . 4 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)):(ℤ × ℕ)–onto→ℚ |
21 | fodomnum 9671 | . . . 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 11836 | . . . . . 6 ⊢ ℕ ∈ V | |
24 | 23 | enref 8661 | . . . . 5 ⊢ ℕ ≈ ℕ |
25 | xpen 8809 | . . . . 5 ⊢ ((ℤ ≈ ℕ ∧ ℕ ≈ ℕ) → (ℤ × ℕ) ≈ (ℕ × ℕ)) | |
26 | 6, 24, 25 | mp2an 692 | . . . 4 ⊢ (ℤ × ℕ) ≈ (ℕ × ℕ) |
27 | xpnnen 15772 | . . . 4 ⊢ (ℕ × ℕ) ≈ ℕ | |
28 | 26, 27 | entri 8682 | . . 3 ⊢ (ℤ × ℕ) ≈ ℕ |
29 | domentr 8687 | . . 3 ⊢ ((ℚ ≼ (ℤ × ℕ) ∧ (ℤ × ℕ) ≈ ℕ) → ℚ ≼ ℕ) | |
30 | 22, 28, 29 | mp2an 692 | . 2 ⊢ ℚ ≼ ℕ |
31 | qex 12557 | . . 3 ⊢ ℚ ∈ V | |
32 | nnssq 12554 | . . 3 ⊢ ℕ ⊆ ℚ | |
33 | ssdomg 8674 | . . 3 ⊢ (ℚ ∈ V → (ℕ ⊆ ℚ → ℕ ≼ ℚ)) | |
34 | 31, 32, 33 | mp2 9 | . 2 ⊢ ℕ ≼ ℚ |
35 | sbth 8766 | . 2 ⊢ ((ℚ ≼ ℕ ∧ ℕ ≼ ℚ) → ℚ ≈ ℕ) | |
36 | 30, 34, 35 | mp2an 692 | 1 ⊢ ℚ ≈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ∈ wcel 2110 {cab 2714 ∃wrex 3062 Vcvv 3408 ⊆ wss 3866 class class class wbr 5053 × cxp 5549 dom cdm 5551 ran crn 5552 Oncon0 6213 Fn wfn 6375 –onto→wfo 6378 (class class class)co 7213 ∈ cmpo 7215 ωcom 7644 ≈ cen 8623 ≼ cdom 8624 cardccrd 9551 / cdiv 11489 ℕcn 11830 ℤcz 12176 ℚcq 12544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-oadd 8206 df-omul 8207 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-oi 9126 df-card 9555 df-acn 9558 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-n0 12091 df-z 12177 df-uz 12439 df-q 12545 |
This theorem is referenced by: rpnnen 15788 resdomq 15805 re2ndc 23698 ovolq 24388 opnmblALT 24500 vitali 24510 mbfimaopnlem 24552 mbfaddlem 24557 mblfinlem1 35551 irrapx1 40353 qenom 42573 |
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