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Theorem qnnen 15558
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 9101 . . . . . . 7 ω ∈ On
2 nnenom 13341 . . . . . . . 8 ℕ ≈ ω
32ensymi 8551 . . . . . . 7 ω ≈ ℕ
4 isnumi 9367 . . . . . . 7 ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
51, 3, 4mp2an 688 . . . . . 6 ℕ ∈ dom card
6 znnen 15557 . . . . . . 7 ℤ ≈ ℕ
7 ennum 9368 . . . . . . 7 (ℤ ≈ ℕ → (ℤ ∈ dom card ↔ ℕ ∈ dom card))
86, 7ax-mp 5 . . . . . 6 (ℤ ∈ dom card ↔ ℕ ∈ dom card)
95, 8mpbir 232 . . . . 5 ℤ ∈ dom card
10 xpnum 9372 . . . . 5 ((ℤ ∈ dom card ∧ ℕ ∈ dom card) → (ℤ × ℕ) ∈ dom card)
119, 5, 10mp2an 688 . . . 4 (ℤ × ℕ) ∈ dom card
12 eqid 2824 . . . . . 6 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦))
13 ovex 7184 . . . . . 6 (𝑥 / 𝑦) ∈ V
1412, 13fnmpoi 7762 . . . . 5 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) Fn (ℤ × ℕ)
1512rnmpo 7277 . . . . . 6 ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)}
16 elq 12342 . . . . . . 7 (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦))
1716abbi2i 2957 . . . . . 6 ℚ = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)}
1815, 17eqtr4i 2851 . . . . 5 ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = ℚ
19 df-fo 6357 . . . . 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 707 . . . 4 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)):(ℤ × ℕ)–onto→ℚ
21 fodomnum 9475 . . . 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 11636 . . . . . 6 ℕ ∈ V
2423enref 8534 . . . . 5 ℕ ≈ ℕ
25 xpen 8672 . . . . 5 ((ℤ ≈ ℕ ∧ ℕ ≈ ℕ) → (ℤ × ℕ) ≈ (ℕ × ℕ))
266, 24, 25mp2an 688 . . . 4 (ℤ × ℕ) ≈ (ℕ × ℕ)
27 xpnnen 15556 . . . 4 (ℕ × ℕ) ≈ ℕ
2826, 27entri 8555 . . 3 (ℤ × ℕ) ≈ ℕ
29 domentr 8560 . . 3 ((ℚ ≼ (ℤ × ℕ) ∧ (ℤ × ℕ) ≈ ℕ) → ℚ ≼ ℕ)
3022, 28, 29mp2an 688 . 2 ℚ ≼ ℕ
31 qex 12353 . . 3 ℚ ∈ V
32 nnssq 12350 . . 3 ℕ ⊆ ℚ
33 ssdomg 8547 . . 3 (ℚ ∈ V → (ℕ ⊆ ℚ → ℕ ≼ ℚ))
3431, 32, 33mp2 9 . 2 ℕ ≼ ℚ
35 sbth 8629 . 2 ((ℚ ≼ ℕ ∧ ℕ ≼ ℚ) → ℚ ≈ ℕ)
3630, 34, 35mp2an 688 1 ℚ ≈ ℕ
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1530  wcel 2106  {cab 2802  wrex 3143  Vcvv 3499  wss 3939   class class class wbr 5062   × cxp 5551  dom cdm 5553  ran crn 5554  Oncon0 6188   Fn wfn 6346  ontowfo 6349  (class class class)co 7151  cmpo 7153  ωcom 7571  cen 8498  cdom 8499  cardccrd 9356   / cdiv 11289  cn 11630  cz 11973  cq 12340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-omul 8101  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-oi 8966  df-card 9360  df-acn 9363  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341
This theorem is referenced by:  rpnnen  15572  resdomq  15589  re2ndc  23324  ovolq  24007  opnmblALT  24119  vitali  24129  mbfimaopnlem  24171  mbfaddlem  24176  mblfinlem1  34797  irrapx1  39287  qenom  41491
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