Theorem qnnen 16171

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.

Step	Hyp	Ref	Expression
1		omelon 9558	. . . . . . 7 ⊢ ω ∈ On
2		nnenom 13933	. . . . . . . 8 ⊢ ℕ ≈ ω
3	2	ensymi 8944	. . . . . . 7 ⊢ ω ≈ ℕ
4		isnumi 9861	. . . . . . 7 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
5	1, 3, 4	mp2an 693	. . . . . 6 ⊢ ℕ ∈ dom card
6		znnen 16170	. . . . . . 7 ⊢ ℤ ≈ ℕ
7		ennum 9862	. . . . . . 7 ⊢ (ℤ ≈ ℕ → (ℤ ∈ dom card ↔ ℕ ∈ dom card))
8	6, 7	ax-mp 5	. . . . . 6 ⊢ (ℤ ∈ dom card ↔ ℕ ∈ dom card)
9	5, 8	mpbir 231	. . . . 5 ⊢ ℤ ∈ dom card
10		xpnum 9866	. . . . 5 ⊢ ((ℤ ∈ dom card ∧ ℕ ∈ dom card) → (ℤ × ℕ) ∈ dom card)
11	9, 5, 10	mp2an 693	. . . 4 ⊢ (ℤ × ℕ) ∈ dom card
12		eqid 2737	. . . . . 6 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦))
13		ovex 7393	. . . . . 6 ⊢ (𝑥 / 𝑦) ∈ V
14	12, 13	fnmpoi 8016	. . . . 5 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) Fn (ℤ × ℕ)
15	12	rnmpo 7493	. . . . . 6 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)}
16		elq 12891	. . . . . . 7 ⊢ (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦))
17	16	eqabi 2872	. . . . . 6 ⊢ ℚ = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)}
18	15, 17	eqtr4i 2763	. . . . 5 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = ℚ
19		df-fo 6498	. . . . 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))
20	14, 18, 19	mpbir2an 712	. . . 4 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)):(ℤ × ℕ)–onto→ℚ
21		fodomnum 9970	. . . 4 ⊢ ((ℤ × ℕ) ∈ dom card → ((𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)):(ℤ × ℕ)–onto→ℚ → ℚ ≼ (ℤ × ℕ)))
22	11, 20, 21	mp2 9	. . 3 ⊢ ℚ ≼ (ℤ × ℕ)
23		nnex 12171	. . . . . 6 ⊢ ℕ ∈ V
24	23	enref 8925	. . . . 5 ⊢ ℕ ≈ ℕ
25		xpen 9071	. . . . 5 ⊢ ((ℤ ≈ ℕ ∧ ℕ ≈ ℕ) → (ℤ × ℕ) ≈ (ℕ × ℕ))
26	6, 24, 25	mp2an 693	. . . 4 ⊢ (ℤ × ℕ) ≈ (ℕ × ℕ)
27		xpnnen 16169	. . . 4 ⊢ (ℕ × ℕ) ≈ ℕ
28	26, 27	entri 8948	. . 3 ⊢ (ℤ × ℕ) ≈ ℕ
29		domentr 8953	. . 3 ⊢ ((ℚ ≼ (ℤ × ℕ) ∧ (ℤ × ℕ) ≈ ℕ) → ℚ ≼ ℕ)
30	22, 28, 29	mp2an 693	. 2 ⊢ ℚ ≼ ℕ
31		qex 12902	. . 3 ⊢ ℚ ∈ V
32		nnssq 12899	. . 3 ⊢ ℕ ⊆ ℚ
33		ssdomg 8940	. . 3 ⊢ (ℚ ∈ V → (ℕ ⊆ ℚ → ℕ ≼ ℚ))
34	31, 32, 33	mp2 9	. 2 ⊢ ℕ ≼ ℚ
35		sbth 9028	. 2 ⊢ ((ℚ ≼ ℕ ∧ ℕ ≼ ℚ) → ℚ ≈ ℕ)
36	30, 34, 35	mp2an 693	1 ⊢ ℚ ≈ ℕ

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062  Vcvv 3430   ⊆ wss 3890   class class class wbr 5086   × cxp 5622  dom cdm 5624  ran crn 5625  Oncon0 6317   Fn wfn 6487  –onto→wfo 6490  (class class class)co 7360   ∈ cmpo 7362  ωcom 7810   ≈ cen 8883   ≼ cdom 8884  cardccrd 9850   / cdiv 11798  ℕcn 12165  ℤcz 12515  ℚcq 12889

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-inf2 9553  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-oadd 8402  df-omul 8403  df-er 8636  df-map 8768  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-oi 9418  df-card 9854  df-acn 9857  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-n0 12429  df-z 12516  df-uz 12780  df-q 12890

This theorem is referenced by:  rpnnen  16185  resdomq  16202  ex-chn2  18595  re2ndc  24776  ovolq  25468  opnmblALT  25580  vitali  25590  mbfimaopnlem  25632  mbfaddlem  25637  mblfinlem1  37992  irrapx1  43274  qenom  45809