Theorem rp-isfinite6 43480

Description: A set is said to be finite if it is either empty or it can be put in one-to-one correspondence with all the natural numbers between 1 and some 𝑛 ∈ ℕ. (Contributed by RP, 10-Mar-2020.)

Assertion

Ref	Expression
rp-isfinite6	⊢ (𝐴 ∈ Fin ↔ (𝐴 = ∅ ∨ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴))

Distinct variable group:   𝐴,𝑛

Proof of Theorem rp-isfinite6

Step	Hyp	Ref	Expression
1		exmid 893	. . . 4 ⊢ (𝐴 = ∅ ∨ ¬ 𝐴 = ∅)
2	1	biantrur 530	. . 3 ⊢ (𝐴 ∈ Fin ↔ ((𝐴 = ∅ ∨ ¬ 𝐴 = ∅) ∧ 𝐴 ∈ Fin))
3		andir 1009	. . 3 ⊢ (((𝐴 = ∅ ∨ ¬ 𝐴 = ∅) ∧ 𝐴 ∈ Fin) ↔ ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ∨ (¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin)))
4	2, 3	bitri 275	. 2 ⊢ (𝐴 ∈ Fin ↔ ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ∨ (¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin)))
5		simpl 482	. . . 4 ⊢ ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) → 𝐴 = ∅)
6		0fi 9108	. . . . . 6 ⊢ ∅ ∈ Fin
7		eleq1a 2839	. . . . . 6 ⊢ (∅ ∈ Fin → (𝐴 = ∅ → 𝐴 ∈ Fin))
8	6, 7	ax-mp 5	. . . . 5 ⊢ (𝐴 = ∅ → 𝐴 ∈ Fin)
9	8	ancli 548	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 = ∅ ∧ 𝐴 ∈ Fin))
10	5, 9	impbii 209	. . 3 ⊢ ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ↔ 𝐴 = ∅)
11		rp-isfinite5 43479	. . . . . 6 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴)
12		df-rex 3077	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴 ↔ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴))
13	11, 12	bitri 275	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴))
14	13	anbi2i 622	. . . 4 ⊢ ((¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin) ↔ (¬ 𝐴 = ∅ ∧ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)))
15		df-rex 3077	. . . . 5 ⊢ (∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴 ↔ ∃𝑛(𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
16		en0 9078	. . . . . . . . . . . . . 14 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
17		ensymb 9062	. . . . . . . . . . . . . 14 ⊢ (𝐴 ≈ ∅ ↔ ∅ ≈ 𝐴)
18	16, 17	bitr3i 277	. . . . . . . . . . . . 13 ⊢ (𝐴 = ∅ ↔ ∅ ≈ 𝐴)
19	18	notbii 320	. . . . . . . . . . . 12 ⊢ (¬ 𝐴 = ∅ ↔ ¬ ∅ ≈ 𝐴)
20		elnn0 12555	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ0 ↔ (𝑛 ∈ ℕ ∨ 𝑛 = 0))
21	20	anbi1i 623	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) ↔ ((𝑛 ∈ ℕ ∨ 𝑛 = 0) ∧ (1...𝑛) ≈ 𝐴))
22		andir 1009	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ ∨ 𝑛 = 0) ∧ (1...𝑛) ≈ 𝐴) ↔ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)))
23	21, 22	bitri 275	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) ↔ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)))
24	19, 23	anbi12i 627	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ (¬ ∅ ≈ 𝐴 ∧ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))))
25		andi 1008	. . . . . . . . . . 11 ⊢ ((¬ ∅ ≈ 𝐴 ∧ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))) ↔ ((¬ ∅ ≈ 𝐴 ∧ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴)) ∨ (¬ ∅ ≈ 𝐴 ∧ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))))
26	24, 25	bitri 275	. . . . . . . . . 10 ⊢ ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ ((¬ ∅ ≈ 𝐴 ∧ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴)) ∨ (¬ ∅ ≈ 𝐴 ∧ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))))
27		3anass 1095	. . . . . . . . . . 11 ⊢ ((¬ ∅ ≈ 𝐴 ∧ 𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ↔ (¬ ∅ ≈ 𝐴 ∧ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴)))
28		3anass 1095	. . . . . . . . . . 11 ⊢ ((¬ ∅ ≈ 𝐴 ∧ 𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴) ↔ (¬ ∅ ≈ 𝐴 ∧ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)))
29	27, 28	orbi12i 913	. . . . . . . . . 10 ⊢ (((¬ ∅ ≈ 𝐴 ∧ 𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (¬ ∅ ≈ 𝐴 ∧ 𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)) ↔ ((¬ ∅ ≈ 𝐴 ∧ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴)) ∨ (¬ ∅ ≈ 𝐴 ∧ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))))
30	26, 29	sylbb2 238	. . . . . . . . 9 ⊢ ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) → ((¬ ∅ ≈ 𝐴 ∧ 𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (¬ ∅ ≈ 𝐴 ∧ 𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)))
31		simp2 1137	. . . . . . . . . 10 ⊢ ((¬ ∅ ≈ 𝐴 ∧ 𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → 𝑛 ∈ ℕ)
32		oveq2 7456	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → (1...𝑛) = (1...0))
33		fz10 13605	. . . . . . . . . . . 12 ⊢ (1...0) = ∅
34	32, 33	eqtrdi 2796	. . . . . . . . . . 11 ⊢ (𝑛 = 0 → (1...𝑛) = ∅)
35		simp2 1137	. . . . . . . . . . . . 13 ⊢ ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → (1...𝑛) = ∅)
36		simp3 1138	. . . . . . . . . . . . 13 ⊢ ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → (1...𝑛) ≈ 𝐴)
37	35, 36	eqbrtrrd 5190	. . . . . . . . . . . 12 ⊢ ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → ∅ ≈ 𝐴)
38		simp1 1136	. . . . . . . . . . . 12 ⊢ ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → ¬ ∅ ≈ 𝐴)
39	37, 38	pm2.21dd 195	. . . . . . . . . . 11 ⊢ ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → 𝑛 ∈ ℕ)
40	34, 39	syl3an2 1164	. . . . . . . . . 10 ⊢ ((¬ ∅ ≈ 𝐴 ∧ 𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴) → 𝑛 ∈ ℕ)
41	31, 40	jaoi 856	. . . . . . . . 9 ⊢ (((¬ ∅ ≈ 𝐴 ∧ 𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (¬ ∅ ≈ 𝐴 ∧ 𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)) → 𝑛 ∈ ℕ)
42	30, 41	syl 17	. . . . . . . 8 ⊢ ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) → 𝑛 ∈ ℕ)
43		simprr 772	. . . . . . . 8 ⊢ ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) → (1...𝑛) ≈ 𝐴)
44	42, 43	jca 511	. . . . . . 7 ⊢ ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) → (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
45		nngt0 12324	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 < 𝑛)
46		hash0 14416	. . . . . . . . . . . . 13 ⊢ (♯‘∅) = 0
47	46	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (♯‘∅) = 0)
48		nnnn0 12560	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
49		hashfz1 14395	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ0 → (♯‘(1...𝑛)) = 𝑛)
50	48, 49	syl 17	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (♯‘(1...𝑛)) = 𝑛)
51	45, 47, 50	3brtr4d 5198	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (♯‘∅) < (♯‘(1...𝑛)))
52		fzfi 14023	. . . . . . . . . . . 12 ⊢ (1...𝑛) ∈ Fin
53		hashsdom 14430	. . . . . . . . . . . 12 ⊢ ((∅ ∈ Fin ∧ (1...𝑛) ∈ Fin) → ((♯‘∅) < (♯‘(1...𝑛)) ↔ ∅ ≺ (1...𝑛)))
54	6, 52, 53	mp2an 691	. . . . . . . . . . 11 ⊢ ((♯‘∅) < (♯‘(1...𝑛)) ↔ ∅ ≺ (1...𝑛))
55	51, 54	sylib 218	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ∅ ≺ (1...𝑛))
56	55	anim1i 614	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → (∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴))
57		sdomentr 9177	. . . . . . . . . . 11 ⊢ ((∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → ∅ ≺ 𝐴)
58		sdomnen 9041	. . . . . . . . . . 11 ⊢ (∅ ≺ 𝐴 → ¬ ∅ ≈ 𝐴)
59	57, 58	syl 17	. . . . . . . . . 10 ⊢ ((∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → ¬ ∅ ≈ 𝐴)
60		en0r 9081	. . . . . . . . . . 11 ⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅)
61	60	notbii 320	. . . . . . . . . 10 ⊢ (¬ ∅ ≈ 𝐴 ↔ ¬ 𝐴 = ∅)
62	59, 61	sylib 218	. . . . . . . . 9 ⊢ ((∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → ¬ 𝐴 = ∅)
63	56, 62	syl 17	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → ¬ 𝐴 = ∅)
64	48	anim1i 614	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴))
65	63, 64	jca 511	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → (¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)))
66	44, 65	impbii 209	. . . . . 6 ⊢ ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
67	66	exbii 1846	. . . . 5 ⊢ (∃𝑛(¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ ∃𝑛(𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
68		19.42v 1953	. . . . 5 ⊢ (∃𝑛(¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ (¬ 𝐴 = ∅ ∧ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)))
69	15, 67, 68	3bitr2ri 300	. . . 4 ⊢ ((¬ 𝐴 = ∅ ∧ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴)
70	14, 69	bitri 275	. . 3 ⊢ ((¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin) ↔ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴)
71	10, 70	orbi12i 913	. 2 ⊢ (((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ∨ (¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin)) ↔ (𝐴 = ∅ ∨ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴))
72	4, 71	bitri 275	1 ⊢ (𝐴 ∈ Fin ↔ (𝐴 = ∅ ∨ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∃wrex 3076  ∅c0 4352   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448   ≈ cen 9000   ≺ csdm 9002  Fincfn 9003  0cc0 11184  1c1 11185   < clt 11324  ℕcn 12293  ℕ₀cn0 12553  ...cfz 13567  ♯chash 14379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-oadd 8526  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-xnn0 12626  df-z 12640  df-uz 12904  df-fz 13568  df-hash 14380

This theorem is referenced by: (None)