Theorem qnnen 16140

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 9561	. . . . . . 7 ⊢ ω ∈ On
2		nnenom 13905	. . . . . . . 8 ⊢ ℕ ≈ ω
3	2	ensymi 8936	. . . . . . 7 ⊢ ω ≈ ℕ
4		isnumi 9861	. . . . . . 7 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
5	1, 3, 4	mp2an 692	. . . . . 6 ⊢ ℕ ∈ dom card
6		znnen 16139	. . . . . . 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 692	. . . 4 ⊢ (ℤ × ℕ) ∈ dom card
12		eqid 2729	. . . . . 6 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦))
13		ovex 7386	. . . . . 6 ⊢ (𝑥 / 𝑦) ∈ V
14	12, 13	fnmpoi 8012	. . . . 5 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) Fn (ℤ × ℕ)
15	12	rnmpo 7486	. . . . . 6 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)}
16		elq 12869	. . . . . . 7 ⊢ (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦))
17	16	eqabi 2863	. . . . . 6 ⊢ ℚ = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)}
18	15, 17	eqtr4i 2755	. . . . 5 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = ℚ
19		df-fo 6492	. . . . 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 711	. . . 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 12152	. . . . . 6 ⊢ ℕ ∈ V
24	23	enref 8917	. . . . 5 ⊢ ℕ ≈ ℕ
25		xpen 9064	. . . . 5 ⊢ ((ℤ ≈ ℕ ∧ ℕ ≈ ℕ) → (ℤ × ℕ) ≈ (ℕ × ℕ))
26	6, 24, 25	mp2an 692	. . . 4 ⊢ (ℤ × ℕ) ≈ (ℕ × ℕ)
27		xpnnen 16138	. . . 4 ⊢ (ℕ × ℕ) ≈ ℕ
28	26, 27	entri 8940	. . 3 ⊢ (ℤ × ℕ) ≈ ℕ
29		domentr 8945	. . 3 ⊢ ((ℚ ≼ (ℤ × ℕ) ∧ (ℤ × ℕ) ≈ ℕ) → ℚ ≼ ℕ)
30	22, 28, 29	mp2an 692	. 2 ⊢ ℚ ≼ ℕ
31		qex 12880	. . 3 ⊢ ℚ ∈ V
32		nnssq 12877	. . 3 ⊢ ℕ ⊆ ℚ
33		ssdomg 8932	. . 3 ⊢ (ℚ ∈ V → (ℕ ⊆ ℚ → ℕ ≼ ℚ))
34	31, 32, 33	mp2 9	. 2 ⊢ ℕ ≼ ℚ
35		sbth 9021	. 2 ⊢ ((ℚ ≼ ℕ ∧ ℕ ≼ ℚ) → ℚ ≈ ℕ)
36	30, 34, 35	mp2an 692	1 ⊢ ℚ ≈ ℕ

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053  Vcvv 3438   ⊆ wss 3905   class class class wbr 5095   × cxp 5621  dom cdm 5623  ran crn 5624  Oncon0 6311   Fn wfn 6481  –onto→wfo 6484  (class class class)co 7353   ∈ cmpo 7355  ωcom 7806   ≈ cen 8876   ≼ cdom 8877  cardccrd 9850   / cdiv 11795  ℕcn 12146  ℤcz 12489  ℚcq 12867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-inf2 9556  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-omul 8400  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-oi 9421  df-card 9854  df-acn 9857  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-div 11796  df-nn 12147  df-n0 12403  df-z 12490  df-uz 12754  df-q 12868

This theorem is referenced by:  rpnnen  16154  resdomq  16171  re2ndc  24705  ovolq  25408  opnmblALT  25520  vitali  25530  mbfimaopnlem  25572  mbfaddlem  25577  mblfinlem1  37639  irrapx1  42804  qenom  45344