MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  qnnen Structured version   Visualization version   GIF version

Theorem qnnen 16246
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 9684 . . . . . . 7 ω ∈ On
2 nnenom 14018 . . . . . . . 8 ℕ ≈ ω
32ensymi 9043 . . . . . . 7 ω ≈ ℕ
4 isnumi 9984 . . . . . . 7 ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
51, 3, 4mp2an 692 . . . . . 6 ℕ ∈ dom card
6 znnen 16245 . . . . . . 7 ℤ ≈ ℕ
7 ennum 9985 . . . . . . 7 (ℤ ≈ ℕ → (ℤ ∈ dom card ↔ ℕ ∈ dom card))
86, 7ax-mp 5 . . . . . 6 (ℤ ∈ dom card ↔ ℕ ∈ dom card)
95, 8mpbir 231 . . . . 5 ℤ ∈ dom card
10 xpnum 9989 . . . . 5 ((ℤ ∈ dom card ∧ ℕ ∈ dom card) → (ℤ × ℕ) ∈ dom card)
119, 5, 10mp2an 692 . . . 4 (ℤ × ℕ) ∈ dom card
12 eqid 2735 . . . . . 6 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦))
13 ovex 7464 . . . . . 6 (𝑥 / 𝑦) ∈ V
1412, 13fnmpoi 8094 . . . . 5 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) Fn (ℤ × ℕ)
1512rnmpo 7566 . . . . . 6 ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)}
16 elq 12990 . . . . . . 7 (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦))
1716eqabi 2875 . . . . . 6 ℚ = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)}
1815, 17eqtr4i 2766 . . . . 5 ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = ℚ
19 df-fo 6569 . . . . 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 711 . . . 4 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)):(ℤ × ℕ)–onto→ℚ
21 fodomnum 10095 . . . 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 12270 . . . . . 6 ℕ ∈ V
2423enref 9024 . . . . 5 ℕ ≈ ℕ
25 xpen 9179 . . . . 5 ((ℤ ≈ ℕ ∧ ℕ ≈ ℕ) → (ℤ × ℕ) ≈ (ℕ × ℕ))
266, 24, 25mp2an 692 . . . 4 (ℤ × ℕ) ≈ (ℕ × ℕ)
27 xpnnen 16244 . . . 4 (ℕ × ℕ) ≈ ℕ
2826, 27entri 9047 . . 3 (ℤ × ℕ) ≈ ℕ
29 domentr 9052 . . 3 ((ℚ ≼ (ℤ × ℕ) ∧ (ℤ × ℕ) ≈ ℕ) → ℚ ≼ ℕ)
3022, 28, 29mp2an 692 . 2 ℚ ≼ ℕ
31 qex 13001 . . 3 ℚ ∈ V
32 nnssq 12998 . . 3 ℕ ⊆ ℚ
33 ssdomg 9039 . . 3 (ℚ ∈ V → (ℕ ⊆ ℚ → ℕ ≼ ℚ))
3431, 32, 33mp2 9 . 2 ℕ ≼ ℚ
35 sbth 9132 . 2 ((ℚ ≼ ℕ ∧ ℕ ≼ ℚ) → ℚ ≈ ℕ)
3630, 34, 35mp2an 692 1 ℚ ≈ ℕ
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wcel 2106  {cab 2712  wrex 3068  Vcvv 3478  wss 3963   class class class wbr 5148   × cxp 5687  dom cdm 5689  ran crn 5690  Oncon0 6386   Fn wfn 6558  ontowfo 6561  (class class class)co 7431  cmpo 7433  ωcom 7887  cen 8981  cdom 8982  cardccrd 9973   / cdiv 11918  cn 12264  cz 12611  cq 12988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-oi 9548  df-card 9977  df-acn 9980  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989
This theorem is referenced by:  rpnnen  16260  resdomq  16277  re2ndc  24837  ovolq  25540  opnmblALT  25652  vitali  25662  mbfimaopnlem  25704  mbfaddlem  25709  mblfinlem1  37644  irrapx1  42816  qenom  45311
  Copyright terms: Public domain W3C validator