Theorem qnnen 16270

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 9616	. . . . . . 7 ⊢ ω ∈ On
2		nnenom 14018	. . . . . . . 8 ⊢ ℕ ≈ ω
3	2	ensymi 9002	. . . . . . 7 ⊢ ω ≈ ℕ
4		isnumi 9933	. . . . . . 7 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
5	1, 3, 4	mp2an 704	. . . . . 6 ⊢ ℕ ∈ dom card
6		znnen 16269	. . . . . . 7 ⊢ ℤ ≈ ℕ
7		ennum 9934	. . . . . . 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 234	. . . . 5 ⊢ ℤ ∈ dom card
10		xpnum 9938	. . . . 5 ⊢ ((ℤ ∈ dom card ∧ ℕ ∈ dom card) → (ℤ × ℕ) ∈ dom card)
11	9, 5, 10	mp2an 704	. . . 4 ⊢ (ℤ × ℕ) ∈ dom card
12		eqid 2763	. . . . . 6 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦))
13		ovex 7445	. . . . . 6 ⊢ (𝑥 / 𝑦) ∈ V
14	12, 13	fnmpoi 8068	. . . . 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 12975	. . . . . . 7 ⊢ (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦))
17	16	eqabi 2898	. . . . . 6 ⊢ ℚ = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦)}
18	15, 17	eqtr4i 2789	. . . . 5 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)) = ℚ
19		df-fo 6544	. . . . 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 723	. . . 4 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ ↦ (𝑥 / 𝑦)):(ℤ × ℕ)–onto→ℚ
21		fodomnum 10042	. . . 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 12240	. . . . . 6 ⊢ ℕ ∈ V
24	23	enref 8983	. . . . 5 ⊢ ℕ ≈ ℕ
25		xpen 9129	. . . . 5 ⊢ ((ℤ ≈ ℕ ∧ ℕ ≈ ℕ) → (ℤ × ℕ) ≈ (ℕ × ℕ))
26	6, 24, 25	mp2an 704	. . . 4 ⊢ (ℤ × ℕ) ≈ (ℕ × ℕ)
27		xpnnen 16268	. . . 4 ⊢ (ℕ × ℕ) ≈ ℕ
28	26, 27	entri 9006	. . 3 ⊢ (ℤ × ℕ) ≈ ℕ
29		domentr 9011	. . 3 ⊢ ((ℚ ≼ (ℤ × ℕ) ∧ (ℤ × ℕ) ≈ ℕ) → ℚ ≼ ℕ)
30	22, 28, 29	mp2an 704	. 2 ⊢ ℚ ≼ ℕ
31		qex 12986	. . 3 ⊢ ℚ ∈ V
32		nnssq 12983	. . 3 ⊢ ℕ ⊆ ℚ
33		ssdomg 8998	. . 3 ⊢ (ℚ ∈ V → (ℕ ⊆ ℚ → ℕ ≼ ℚ))
34	31, 32, 33	mp2 9	. 2 ⊢ ℕ ≼ ℚ
35		sbth 9086	. 2 ⊢ ((ℚ ≼ ℕ ∧ ℕ ≼ ℚ) → ℚ ≈ ℕ)
36	30, 34, 35	mp2an 704	1 ⊢ ℚ ≈ ℕ

Syntax hints:   ↔ wb 209   = wceq 1570   ∈ wcel 2143  {cab 2741  ∃wrex 3089  Vcvv 3455   ⊆ wss 3906   class class class wbr 5110   × cxp 5661  dom cdm 5663  ran crn 5664  Oncon0 6362   Fn wfn 6533  –onto→wfo 6536  (class class class)co 7412   ∈ cmpo 7414  ωcom 7863   ≈ cen 8941   ≼ cdom 8942  cardccrd 9922   / cdiv 11872  ℕcn 12234  ℤcz 12592  ℚcq 12973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-inf2 9611  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-oadd 8458  df-omul 8459  df-er 8695  df-map 8827  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-oi 9473  df-card 9926  df-acn 9929  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-div 11873  df-nn 12235  df-n0 12506  df-z 12593  df-uz 12864  df-q 12974

This theorem is referenced by:  rpnnen  16284  resdomq  16301  ex-chn2  18695  re2ndc  24939  ovolq  25631  opnmblALT  25743  vitali  25753  mbfimaopnlem  25795  mbfaddlem  25800  mblfinlem1  38289  irrapx1  43538  qenom  46060