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Theorem qnnen 15561
 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 9096 . . . . . . 7 ω ∈ On
2 nnenom 13346 . . . . . . . 8 ℕ ≈ ω
32ensymi 8545 . . . . . . 7 ω ≈ ℕ
4 isnumi 9362 . . . . . . 7 ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
51, 3, 4mp2an 691 . . . . . 6 ℕ ∈ dom card
6 znnen 15560 . . . . . . 7 ℤ ≈ ℕ
7 ennum 9363 . . . . . . 7 (ℤ ≈ ℕ → (ℤ ∈ dom card ↔ ℕ ∈ dom card))
86, 7ax-mp 5 . . . . . 6 (ℤ ∈ dom card ↔ ℕ ∈ dom card)
95, 8mpbir 234 . . . . 5 ℤ ∈ dom card
10 xpnum 9367 . . . . 5 ((ℤ ∈ dom card ∧ ℕ ∈ dom card) → (ℤ × ℕ) ∈ dom card)
119, 5, 10mp2an 691 . . . 4 (ℤ × ℕ) ∈ dom card
12 eqid 2798 . . . . . 6 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦))
13 ovex 7169 . . . . . 6 (𝑥 / 𝑦) ∈ V
1412, 13fnmpoi 7753 . . . . 5 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) Fn (ℤ × ℕ)
1512rnmpo 7265 . . . . . 6 ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)}
16 elq 12341 . . . . . . 7 (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦))
1716abbi2i 2929 . . . . . 6 ℚ = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)}
1815, 17eqtr4i 2824 . . . . 5 ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = ℚ
19 df-fo 6331 . . . . 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 710 . . . 4 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)):(ℤ × ℕ)–onto→ℚ
21 fodomnum 9471 . . . 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 11634 . . . . . 6 ℕ ∈ V
2423enref 8528 . . . . 5 ℕ ≈ ℕ
25 xpen 8667 . . . . 5 ((ℤ ≈ ℕ ∧ ℕ ≈ ℕ) → (ℤ × ℕ) ≈ (ℕ × ℕ))
266, 24, 25mp2an 691 . . . 4 (ℤ × ℕ) ≈ (ℕ × ℕ)
27 xpnnen 15559 . . . 4 (ℕ × ℕ) ≈ ℕ
2826, 27entri 8549 . . 3 (ℤ × ℕ) ≈ ℕ
29 domentr 8554 . . 3 ((ℚ ≼ (ℤ × ℕ) ∧ (ℤ × ℕ) ≈ ℕ) → ℚ ≼ ℕ)
3022, 28, 29mp2an 691 . 2 ℚ ≼ ℕ
31 qex 12351 . . 3 ℚ ∈ V
32 nnssq 12348 . . 3 ℕ ⊆ ℚ
33 ssdomg 8541 . . 3 (ℚ ∈ V → (ℕ ⊆ ℚ → ℕ ≼ ℚ))
3431, 32, 33mp2 9 . 2 ℕ ≼ ℚ
35 sbth 8624 . 2 ((ℚ ≼ ℕ ∧ ℕ ≼ ℚ) → ℚ ≈ ℕ)
3630, 34, 35mp2an 691 1 ℚ ≈ ℕ
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  {cab 2776  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881   class class class wbr 5031   × cxp 5518  dom cdm 5520  ran crn 5521  Oncon0 6160   Fn wfn 6320  –onto→wfo 6323  (class class class)co 7136   ∈ cmpo 7138  ωcom 7563   ≈ cen 8492   ≼ cdom 8493  cardccrd 9351   / cdiv 11289  ℕcn 11628  ℤcz 11972  ℚcq 12339 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-inf2 9091  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 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-se 5480  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-isom 6334  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-oadd 8092  df-omul 8093  df-er 8275  df-map 8394  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-oi 8961  df-card 9355  df-acn 9358  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 11629  df-n0 11889  df-z 11973  df-uz 12235  df-q 12340 This theorem is referenced by:  rpnnen  15575  resdomq  15592  re2ndc  23416  ovolq  24105  opnmblALT  24217  vitali  24227  mbfimaopnlem  24269  mbfaddlem  24274  mblfinlem1  35113  irrapx1  39812  qenom  42036
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