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Theorem qnnen 15141
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.)
Assertion
Ref Expression
qnnen ℚ ≈ ℕ

Proof of Theorem qnnen
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omelon 8705 . . . . . . 7 ω ∈ On
2 nnenom 12980 . . . . . . . 8 ℕ ≈ ω
32ensymi 8157 . . . . . . 7 ω ≈ ℕ
4 isnumi 8970 . . . . . . 7 ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
51, 3, 4mp2an 672 . . . . . 6 ℕ ∈ dom card
6 znnen 15140 . . . . . . 7 ℤ ≈ ℕ
7 ennum 8971 . . . . . . 7 (ℤ ≈ ℕ → (ℤ ∈ dom card ↔ ℕ ∈ dom card))
86, 7ax-mp 5 . . . . . 6 (ℤ ∈ dom card ↔ ℕ ∈ dom card)
95, 8mpbir 221 . . . . 5 ℤ ∈ dom card
10 xpnum 8975 . . . . 5 ((ℤ ∈ dom card ∧ ℕ ∈ dom card) → (ℤ × ℕ) ∈ dom card)
119, 5, 10mp2an 672 . . . 4 (ℤ × ℕ) ∈ dom card
12 eqid 2771 . . . . . 6 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦))
13 ovex 6821 . . . . . 6 (𝑥 / 𝑦) ∈ V
1412, 13fnmpt2i 7387 . . . . 5 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) Fn (ℤ × ℕ)
1512rnmpt2 6915 . . . . . 6 ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)}
16 elq 11991 . . . . . . 7 (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦))
1716abbi2i 2887 . . . . . 6 ℚ = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)}
1815, 17eqtr4i 2796 . . . . 5 ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = ℚ
19 df-fo 6035 . . . . 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))
2014, 18, 19mpbir2an 690 . . . 4 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)):(ℤ × ℕ)–onto→ℚ
21 fodomnum 9078 . . . 4 ((ℤ × ℕ) ∈ dom card → ((𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)):(ℤ × ℕ)–onto→ℚ → ℚ ≼ (ℤ × ℕ)))
2211, 20, 21mp2 9 . . 3 ℚ ≼ (ℤ × ℕ)
23 nnex 11226 . . . . . 6 ℕ ∈ V
2423enref 8140 . . . . 5 ℕ ≈ ℕ
25 xpen 8277 . . . . 5 ((ℤ ≈ ℕ ∧ ℕ ≈ ℕ) → (ℤ × ℕ) ≈ (ℕ × ℕ))
266, 24, 25mp2an 672 . . . 4 (ℤ × ℕ) ≈ (ℕ × ℕ)
27 xpnnen 15138 . . . 4 (ℕ × ℕ) ≈ ℕ
2826, 27entri 8161 . . 3 (ℤ × ℕ) ≈ ℕ
29 domentr 8166 . . 3 ((ℚ ≼ (ℤ × ℕ) ∧ (ℤ × ℕ) ≈ ℕ) → ℚ ≼ ℕ)
3022, 28, 29mp2an 672 . 2 ℚ ≼ ℕ
31 qex 12001 . . 3 ℚ ∈ V
32 nnssq 11998 . . 3 ℕ ⊆ ℚ
33 ssdomg 8153 . . 3 (ℚ ∈ V → (ℕ ⊆ ℚ → ℕ ≼ ℚ))
3431, 32, 33mp2 9 . 2 ℕ ≼ ℚ
35 sbth 8234 . 2 ((ℚ ≼ ℕ ∧ ℕ ≼ ℚ) → ℚ ≈ ℕ)
3630, 34, 35mp2an 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 5864   Fn wfn 6024  ontowfo 6027  (class class class)co 6791  cmpt2 6793  ωcom 7210  cen 8104  cdom 8105  cardccrd 8959   / cdiv 10884  cn 11220  cz 11577  cq 11989
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 7094  ax-inf2 8700  ax-cnex 10192  ax-resscn 10193  ax-1cn 10194  ax-icn 10195  ax-addcl 10196  ax-addrcl 10197  ax-mulcl 10198  ax-mulrcl 10199  ax-mulcom 10200  ax-addass 10201  ax-mulass 10202  ax-distr 10203  ax-i2m1 10204  ax-1ne0 10205  ax-1rid 10206  ax-rnegex 10207  ax-rrecex 10208  ax-cnre 10209  ax-pre-lttri 10210  ax-pre-lttrn 10211  ax-pre-ltadd 10212  ax-pre-mulgt0 10213
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-isom 6038  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-om 7211  df-1st 7313  df-2nd 7314  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-oadd 7715  df-omul 7716  df-er 7894  df-map 8009  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111  df-oi 8569  df-card 8963  df-acn 8966  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279  df-le 10280  df-sub 10468  df-neg 10469  df-div 10885  df-nn 11221  df-n0 11493  df-z 11578  df-uz 11887  df-q 11990
This theorem is referenced by:  rpnnen  15155  resdomq  15172  re2ndc  22817  ovolq  23472  opnmblALT  23584  vitali  23594  mbfimaopnlem  23635  mbfaddlem  23640  mblfinlem1  33772  irrapx1  37911  qenom  40086
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