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| 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 9687 | . . . . . . 7 ⊢ ω ∈ On | |
| 2 | nnenom 14022 | . . . . . . . 8 ⊢ ℕ ≈ ω | |
| 3 | 2 | ensymi 9045 | . . . . . . 7 ⊢ ω ≈ ℕ | 
| 4 | isnumi 9987 | . . . . . . 7 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card) | |
| 5 | 1, 3, 4 | mp2an 692 | . . . . . 6 ⊢ ℕ ∈ dom card | 
| 6 | znnen 16249 | . . . . . . 7 ⊢ ℤ ≈ ℕ | |
| 7 | ennum 9988 | . . . . . . 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 231 | . . . . 5 ⊢ ℤ ∈ dom card | 
| 10 | xpnum 9992 | . . . . 5 ⊢ ((ℤ ∈ dom card ∧ ℕ ∈ dom card) → (ℤ × ℕ) ∈ dom card) | |
| 11 | 9, 5, 10 | mp2an 692 | . . . 4 ⊢ (ℤ × ℕ) ∈ dom card | 
| 12 | eqid 2736 | . . . . . 6 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) | |
| 13 | ovex 7465 | . . . . . 6 ⊢ (𝑥 / 𝑦) ∈ V | |
| 14 | 12, 13 | fnmpoi 8096 | . . . . 5 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) Fn (ℤ × ℕ) | 
| 15 | 12 | rnmpo 7567 | . . . . . 6 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)} | 
| 16 | elq 12993 | . . . . . . 7 ⊢ (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)) | |
| 17 | 16 | eqabi 2876 | . . . . . 6 ⊢ ℚ = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)} | 
| 18 | 15, 17 | eqtr4i 2767 | . . . . 5 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = ℚ | 
| 19 | df-fo 6566 | . . . . 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 10098 | . . . 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 12273 | . . . . . 6 ⊢ ℕ ∈ V | |
| 24 | 23 | enref 9026 | . . . . 5 ⊢ ℕ ≈ ℕ | 
| 25 | xpen 9181 | . . . . 5 ⊢ ((ℤ ≈ ℕ ∧ ℕ ≈ ℕ) → (ℤ × ℕ) ≈ (ℕ × ℕ)) | |
| 26 | 6, 24, 25 | mp2an 692 | . . . 4 ⊢ (ℤ × ℕ) ≈ (ℕ × ℕ) | 
| 27 | xpnnen 16248 | . . . 4 ⊢ (ℕ × ℕ) ≈ ℕ | |
| 28 | 26, 27 | entri 9049 | . . 3 ⊢ (ℤ × ℕ) ≈ ℕ | 
| 29 | domentr 9054 | . . 3 ⊢ ((ℚ ≼ (ℤ × ℕ) ∧ (ℤ × ℕ) ≈ ℕ) → ℚ ≼ ℕ) | |
| 30 | 22, 28, 29 | mp2an 692 | . 2 ⊢ ℚ ≼ ℕ | 
| 31 | qex 13004 | . . 3 ⊢ ℚ ∈ V | |
| 32 | nnssq 13001 | . . 3 ⊢ ℕ ⊆ ℚ | |
| 33 | ssdomg 9041 | . . 3 ⊢ (ℚ ∈ V → (ℕ ⊆ ℚ → ℕ ≼ ℚ)) | |
| 34 | 31, 32, 33 | mp2 9 | . 2 ⊢ ℕ ≼ ℚ | 
| 35 | sbth 9134 | . 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 206 = wceq 1539 ∈ wcel 2107 {cab 2713 ∃wrex 3069 Vcvv 3479 ⊆ wss 3950 class class class wbr 5142 × cxp 5682 dom cdm 5684 ran crn 5685 Oncon0 6383 Fn wfn 6555 –onto→wfo 6558 (class class class)co 7432 ∈ cmpo 7434 ωcom 7888 ≈ cen 8983 ≼ cdom 8984 cardccrd 9976 / cdiv 11921 ℕcn 12267 ℤcz 12615 ℚcq 12991 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-oadd 8511 df-omul 8512 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-oi 9551 df-card 9980 df-acn 9983 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-n0 12529 df-z 12616 df-uz 12880 df-q 12992 | 
| This theorem is referenced by: rpnnen 16264 resdomq 16281 re2ndc 24823 ovolq 25527 opnmblALT 25639 vitali 25649 mbfimaopnlem 25691 mbfaddlem 25696 mblfinlem1 37665 irrapx1 42844 qenom 45377 | 
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