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Theorem qnnen 14886
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 8503 . . . . . . 7 ω ∈ On
2 nnenom 12735 . . . . . . . 8 ℕ ≈ ω
32ensymi 7966 . . . . . . 7 ω ≈ ℕ
4 isnumi 8732 . . . . . . 7 ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
51, 3, 4mp2an 707 . . . . . 6 ℕ ∈ dom card
6 znnen 14885 . . . . . . 7 ℤ ≈ ℕ
7 ennum 8733 . . . . . . 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 8737 . . . . 5 ((ℤ ∈ dom card ∧ ℕ ∈ dom card) → (ℤ × ℕ) ∈ dom card)
119, 5, 10mp2an 707 . . . 4 (ℤ × ℕ) ∈ dom card
12 eqid 2621 . . . . . 6 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦))
13 ovex 6643 . . . . . 6 (𝑥 / 𝑦) ∈ V
1412, 13fnmpt2i 7199 . . . . 5 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) Fn (ℤ × ℕ)
1512rnmpt2 6735 . . . . . 6 ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)}
16 elq 11750 . . . . . . 7 (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦))
1716abbi2i 2735 . . . . . 6 ℚ = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)}
1815, 17eqtr4i 2646 . . . . 5 ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = ℚ
19 df-fo 5863 . . . . 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 954 . . . 4 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)):(ℤ × ℕ)–onto→ℚ
21 fodomnum 8840 . . . 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 10986 . . . . . 6 ℕ ∈ V
2423enref 7948 . . . . 5 ℕ ≈ ℕ
25 xpen 8083 . . . . 5 ((ℤ ≈ ℕ ∧ ℕ ≈ ℕ) → (ℤ × ℕ) ≈ (ℕ × ℕ))
266, 24, 25mp2an 707 . . . 4 (ℤ × ℕ) ≈ (ℕ × ℕ)
27 xpnnen 14883 . . . 4 (ℕ × ℕ) ≈ ℕ
2826, 27entri 7970 . . 3 (ℤ × ℕ) ≈ ℕ
29 domentr 7975 . . 3 ((ℚ ≼ (ℤ × ℕ) ∧ (ℤ × ℕ) ≈ ℕ) → ℚ ≼ ℕ)
3022, 28, 29mp2an 707 . 2 ℚ ≼ ℕ
31 qex 11760 . . 3 ℚ ∈ V
32 nnssq 11757 . . 3 ℕ ⊆ ℚ
33 ssdomg 7961 . . 3 (ℚ ∈ V → (ℕ ⊆ ℚ → ℕ ≼ ℚ))
3431, 32, 33mp2 9 . 2 ℕ ≼ ℚ
35 sbth 8040 . 2 ((ℚ ≼ ℕ ∧ ℕ ≼ ℚ) → ℚ ≈ ℕ)
3630, 34, 35mp2an 707 1 ℚ ≈ ℕ
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wcel 1987  {cab 2607  wrex 2909  Vcvv 3190  wss 3560   class class class wbr 4623   × cxp 5082  dom cdm 5084  ran crn 5085  Oncon0 5692   Fn wfn 5852  ontowfo 5855  (class class class)co 6615  cmpt2 6617  ωcom 7027  cen 7912  cdom 7913  cardccrd 8721   / cdiv 10644  cn 10980  cz 11337  cq 11748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-omul 7525  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-oi 8375  df-card 8725  df-acn 8728  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-n0 11253  df-z 11338  df-uz 11648  df-q 11749
This theorem is referenced by:  rpnnen  14900  resdomq  14917  re2ndc  22544  ovolq  23199  opnmblALT  23311  vitali  23322  mbfimaopnlem  23362  mbfaddlem  23367  mblfinlem1  33117  irrapx1  36911  qenom  39076
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