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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 9647 | . . . . . . 7 ⊢ ω ∈ On | |
2 | nnenom 13952 | . . . . . . . 8 ⊢ ℕ ≈ ω | |
3 | 2 | ensymi 9006 | . . . . . . 7 ⊢ ω ≈ ℕ |
4 | isnumi 9947 | . . . . . . 7 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card) | |
5 | 1, 3, 4 | mp2an 689 | . . . . . 6 ⊢ ℕ ∈ dom card |
6 | znnen 16162 | . . . . . . 7 ⊢ ℤ ≈ ℕ | |
7 | ennum 9948 | . . . . . . 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 9952 | . . . . 5 ⊢ ((ℤ ∈ dom card ∧ ℕ ∈ dom card) → (ℤ × ℕ) ∈ dom card) | |
11 | 9, 5, 10 | mp2an 689 | . . . 4 ⊢ (ℤ × ℕ) ∈ dom card |
12 | eqid 2731 | . . . . . 6 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) | |
13 | ovex 7445 | . . . . . 6 ⊢ (𝑥 / 𝑦) ∈ V | |
14 | 12, 13 | fnmpoi 8060 | . . . . 5 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) Fn (ℤ × ℕ) |
15 | 12 | rnmpo 7545 | . . . . . 6 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)} |
16 | elq 12941 | . . . . . . 7 ⊢ (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)) | |
17 | 16 | eqabi 2868 | . . . . . 6 ⊢ ℚ = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)} |
18 | 15, 17 | eqtr4i 2762 | . . . . 5 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = ℚ |
19 | df-fo 6549 | . . . . 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 708 | . . . 4 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)):(ℤ × ℕ)–onto→ℚ |
21 | fodomnum 10058 | . . . 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 12225 | . . . . . 6 ⊢ ℕ ∈ V | |
24 | 23 | enref 8987 | . . . . 5 ⊢ ℕ ≈ ℕ |
25 | xpen 9146 | . . . . 5 ⊢ ((ℤ ≈ ℕ ∧ ℕ ≈ ℕ) → (ℤ × ℕ) ≈ (ℕ × ℕ)) | |
26 | 6, 24, 25 | mp2an 689 | . . . 4 ⊢ (ℤ × ℕ) ≈ (ℕ × ℕ) |
27 | xpnnen 16161 | . . . 4 ⊢ (ℕ × ℕ) ≈ ℕ | |
28 | 26, 27 | entri 9010 | . . 3 ⊢ (ℤ × ℕ) ≈ ℕ |
29 | domentr 9015 | . . 3 ⊢ ((ℚ ≼ (ℤ × ℕ) ∧ (ℤ × ℕ) ≈ ℕ) → ℚ ≼ ℕ) | |
30 | 22, 28, 29 | mp2an 689 | . 2 ⊢ ℚ ≼ ℕ |
31 | qex 12952 | . . 3 ⊢ ℚ ∈ V | |
32 | nnssq 12949 | . . 3 ⊢ ℕ ⊆ ℚ | |
33 | ssdomg 9002 | . . 3 ⊢ (ℚ ∈ V → (ℕ ⊆ ℚ → ℕ ≼ ℚ)) | |
34 | 31, 32, 33 | mp2 9 | . 2 ⊢ ℕ ≼ ℚ |
35 | sbth 9099 | . 2 ⊢ ((ℚ ≼ ℕ ∧ ℕ ≼ ℚ) → ℚ ≈ ℕ) | |
36 | 30, 34, 35 | mp2an 689 | 1 ⊢ ℚ ≈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1540 ∈ wcel 2105 {cab 2708 ∃wrex 3069 Vcvv 3473 ⊆ wss 3948 class class class wbr 5148 × cxp 5674 dom cdm 5676 ran crn 5677 Oncon0 6364 Fn wfn 6538 –onto→wfo 6541 (class class class)co 7412 ∈ cmpo 7414 ωcom 7859 ≈ cen 8942 ≼ cdom 8943 cardccrd 9936 / cdiv 11878 ℕcn 12219 ℤcz 12565 ℚcq 12939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-oadd 8476 df-omul 8477 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-oi 9511 df-card 9940 df-acn 9943 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-n0 12480 df-z 12566 df-uz 12830 df-q 12940 |
This theorem is referenced by: rpnnen 16177 resdomq 16194 re2ndc 24637 ovolq 25340 opnmblALT 25452 vitali 25462 mbfimaopnlem 25504 mbfaddlem 25509 mblfinlem1 36989 irrapx1 42029 qenom 44530 |
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