Theorem qnnen 16236

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 9665	. . . . . . 7 ⊢ ω ∈ On
2		nnenom 14003	. . . . . . . 8 ⊢ ℕ ≈ ω
3	2	ensymi 9023	. . . . . . 7 ⊢ ω ≈ ℕ
4		isnumi 9965	. . . . . . 7 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
5	1, 3, 4	mp2an 692	. . . . . 6 ⊢ ℕ ∈ dom card
6		znnen 16235	. . . . . . 7 ⊢ ℤ ≈ ℕ
7		ennum 9966	. . . . . . 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 9970	. . . . 5 ⊢ ((ℤ ∈ dom card ∧ ℕ ∈ dom card) → (ℤ × ℕ) ∈ dom card)
11	9, 5, 10	mp2an 692	. . . 4 ⊢ (ℤ × ℕ) ∈ dom card
12		eqid 2736	. . . . . 6 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦))
13		ovex 7443	. . . . . 6 ⊢ (𝑥 / 𝑦) ∈ V
14	12, 13	fnmpoi 8074	. . . . 5 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) Fn (ℤ × ℕ)
15	12	rnmpo 7545	. . . . . 6 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)}
16		elq 12971	. . . . . . 7 ⊢ (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦))
17	16	eqabi 2871	. . . . . 6 ⊢ ℚ = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)}
18	15, 17	eqtr4i 2762	. . . . 5 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = ℚ
19		df-fo 6542	. . . . 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 10076	. . . 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 12251	. . . . . 6 ⊢ ℕ ∈ V
24	23	enref 9004	. . . . 5 ⊢ ℕ ≈ ℕ
25		xpen 9159	. . . . 5 ⊢ ((ℤ ≈ ℕ ∧ ℕ ≈ ℕ) → (ℤ × ℕ) ≈ (ℕ × ℕ))
26	6, 24, 25	mp2an 692	. . . 4 ⊢ (ℤ × ℕ) ≈ (ℕ × ℕ)
27		xpnnen 16234	. . . 4 ⊢ (ℕ × ℕ) ≈ ℕ
28	26, 27	entri 9027	. . 3 ⊢ (ℤ × ℕ) ≈ ℕ
29		domentr 9032	. . 3 ⊢ ((ℚ ≼ (ℤ × ℕ) ∧ (ℤ × ℕ) ≈ ℕ) → ℚ ≼ ℕ)
30	22, 28, 29	mp2an 692	. 2 ⊢ ℚ ≼ ℕ
31		qex 12982	. . 3 ⊢ ℚ ∈ V
32		nnssq 12979	. . 3 ⊢ ℕ ⊆ ℚ
33		ssdomg 9019	. . 3 ⊢ (ℚ ∈ V → (ℕ ⊆ ℚ → ℕ ≼ ℚ))
34	31, 32, 33	mp2 9	. 2 ⊢ ℕ ≼ ℚ
35		sbth 9112	. 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 2714  ∃wrex 3061  Vcvv 3464   ⊆ wss 3931   class class class wbr 5124   × cxp 5657  dom cdm 5659  ran crn 5660  Oncon0 6357   Fn wfn 6531  –onto→wfo 6534  (class class class)co 7410   ∈ cmpo 7412  ωcom 7866   ≈ cen 8961   ≼ cdom 8962  cardccrd 9954   / cdiv 11899  ℕcn 12245  ℤcz 12593  ℚcq 12969

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-oadd 8489  df-omul 8490  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-oi 9529  df-card 9958  df-acn 9961  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-div 11900  df-nn 12246  df-n0 12507  df-z 12594  df-uz 12858  df-q 12970

This theorem is referenced by:  rpnnen  16250  resdomq  16267  re2ndc  24745  ovolq  25449  opnmblALT  25561  vitali  25571  mbfimaopnlem  25613  mbfaddlem  25618  mblfinlem1  37686  irrapx1  42818  qenom  45355