Theorem evenennn 13228

Description: There are as many even positive integers as there are positive integers. (Contributed by Jim Kingdon, 12-May-2022.)

Assertion

Ref	Expression
evenennn	⊢ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ≈ ℕ

Proof of Theorem evenennn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnex 9260	. . 3 ⊢ ℕ ∈ V
2	1	rabex 4261	. 2 ⊢ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∈ V
3		breq2 4118	. . . 4 ⊢ (𝑧 = 𝑥 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑥))
4	3	elrab 2976	. . 3 ⊢ (𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ↔ (𝑥 ∈ ℕ ∧ 2 ∥ 𝑥))
5		nnehalf 12615	. . 3 ⊢ ((𝑥 ∈ ℕ ∧ 2 ∥ 𝑥) → (𝑥 / 2) ∈ ℕ)
6	4, 5	sylbi 121	. 2 ⊢ (𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} → (𝑥 / 2) ∈ ℕ)
7		2nn 9416	. . . . 5 ⊢ 2 ∈ ℕ
8	7	a1i 9	. . . 4 ⊢ (𝑦 ∈ ℕ → 2 ∈ ℕ)
9		id 19	. . . 4 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ)
10	8, 9	nnmulcld 9303	. . 3 ⊢ (𝑦 ∈ ℕ → (2 · 𝑦) ∈ ℕ)
11		2z 9622	. . . 4 ⊢ 2 ∈ ℤ
12		nnz 9613	. . . 4 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
13		dvdsmul1 12524	. . . 4 ⊢ ((2 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 2 ∥ (2 · 𝑦))
14	11, 12, 13	sylancr 414	. . 3 ⊢ (𝑦 ∈ ℕ → 2 ∥ (2 · 𝑦))
15		breq2 4118	. . . 4 ⊢ (𝑧 = (2 · 𝑦) → (2 ∥ 𝑧 ↔ 2 ∥ (2 · 𝑦)))
16	15	elrab 2976	. . 3 ⊢ ((2 · 𝑦) ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ↔ ((2 · 𝑦) ∈ ℕ ∧ 2 ∥ (2 · 𝑦)))
17	10, 14, 16	sylanbrc 417	. 2 ⊢ (𝑦 ∈ ℕ → (2 · 𝑦) ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧})
18		elrabi 2973	. . . . . 6 ⊢ (𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} → 𝑥 ∈ ℕ)
19	18	adantr 276	. . . . 5 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 𝑥 ∈ ℕ)
20	19	nncnd 9268	. . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 𝑥 ∈ ℂ)
21		simpr 110	. . . . 5 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℕ)
22	21	nncnd 9268	. . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℂ)
23		2cnd 9327	. . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 2 ∈ ℂ)
24		2ap0 9347	. . . . 5 ⊢ 2 # 0
25	24	a1i 9	. . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 2 # 0)
26	20, 22, 23, 25	divmulap3d 9116	. . 3 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → ((𝑥 / 2) = 𝑦 ↔ 𝑥 = (𝑦 · 2)))
27		eqcom 2236	. . . 4 ⊢ ((𝑥 / 2) = 𝑦 ↔ 𝑦 = (𝑥 / 2))
28	27	a1i 9	. . 3 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → ((𝑥 / 2) = 𝑦 ↔ 𝑦 = (𝑥 / 2)))
29	22, 23	mulcomd 8311	. . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → (𝑦 · 2) = (2 · 𝑦))
30	29	eqeq2d 2246	. . 3 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → (𝑥 = (𝑦 · 2) ↔ 𝑥 = (2 · 𝑦)))
31	26, 28, 30	3bitr3rd 219	. 2 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → (𝑥 = (2 · 𝑦) ↔ 𝑦 = (𝑥 / 2)))
32	2, 1, 6, 17, 31	en3i 7023	1 ⊢ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ≈ ℕ

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  {crab 2526   class class class wbr 4114  (class class class)co 6058   ≈ cen 6986  0cc0 8143   · cmul 8148   # cap 8872   / cdiv 8963  ℕcn 9254  2c2 9305  ℤcz 9594   ∥ cdvds 12498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-po 4422  df-iso 4423  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-en 6989  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-dvds 12499

This theorem is referenced by:  unennn  13232