Theorem rp-isfinite6 43963

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 895	. . . 4 ⊢ (𝐴 = ∅ ∨ ¬ 𝐴 = ∅)
2	1	biantrur 530	. . 3 ⊢ (𝐴 ∈ Fin ↔ ((𝐴 = ∅ ∨ ¬ 𝐴 = ∅) ∧ 𝐴 ∈ Fin))
3		andir 1011	. . 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 8982	. . . . . 6 ⊢ ∅ ∈ Fin
7		eleq1a 2832	. . . . . 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 43962	. . . . . 6 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴)
12		df-rex 3063	. . . . . 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 624	. . . 4 ⊢ ((¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin) ↔ (¬ 𝐴 = ∅ ∧ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)))
15		df-rex 3063	. . . . 5 ⊢ (∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴 ↔ ∃𝑛(𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
16		en0 8958	. . . . . . . . . . . . . 14 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
17		ensymb 8942	. . . . . . . . . . . . . 14 ⊢ (𝐴 ≈ ∅ ↔ ∅ ≈ 𝐴)
18	16, 17	bitr3i 277	. . . . . . . . . . . . 13 ⊢ (𝐴 = ∅ ↔ ∅ ≈ 𝐴)
19	18	notbii 320	. . . . . . . . . . . 12 ⊢ (¬ 𝐴 = ∅ ↔ ¬ ∅ ≈ 𝐴)
20		elnn0 12430	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ0 ↔ (𝑛 ∈ ℕ ∨ 𝑛 = 0))
21	20	anbi1i 625	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) ↔ ((𝑛 ∈ ℕ ∨ 𝑛 = 0) ∧ (1...𝑛) ≈ 𝐴))
22		andir 1011	. . . . . . . . . . . . 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 629	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ (¬ ∅ ≈ 𝐴 ∧ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))))
25		andi 1010	. . . . . . . . . . 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 915	. . . . . . . . . 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 1138	. . . . . . . . . 10 ⊢ ((¬ ∅ ≈ 𝐴 ∧ 𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → 𝑛 ∈ ℕ)
32		oveq2 7368	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → (1...𝑛) = (1...0))
33		fz10 13490	. . . . . . . . . . . 12 ⊢ (1...0) = ∅
34	32, 33	eqtrdi 2788	. . . . . . . . . . 11 ⊢ (𝑛 = 0 → (1...𝑛) = ∅)
35		simp2 1138	. . . . . . . . . . . . 13 ⊢ ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → (1...𝑛) = ∅)
36		simp3 1139	. . . . . . . . . . . . 13 ⊢ ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → (1...𝑛) ≈ 𝐴)
37	35, 36	eqbrtrrd 5110	. . . . . . . . . . . 12 ⊢ ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → ∅ ≈ 𝐴)
38		simp1 1137	. . . . . . . . . . . 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 1165	. . . . . . . . . 10 ⊢ ((¬ ∅ ≈ 𝐴 ∧ 𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴) → 𝑛 ∈ ℕ)
41	31, 40	jaoi 858	. . . . . . . . 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 773	. . . . . . . 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 12199	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 < 𝑛)
46		hash0 14320	. . . . . . . . . . . . 13 ⊢ (♯‘∅) = 0
47	46	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (♯‘∅) = 0)
48		nnnn0 12435	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
49		hashfz1 14299	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ0 → (♯‘(1...𝑛)) = 𝑛)
50	48, 49	syl 17	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (♯‘(1...𝑛)) = 𝑛)
51	45, 47, 50	3brtr4d 5118	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (♯‘∅) < (♯‘(1...𝑛)))
52		fzfi 13925	. . . . . . . . . . . 12 ⊢ (1...𝑛) ∈ Fin
53		hashsdom 14334	. . . . . . . . . . . 12 ⊢ ((∅ ∈ Fin ∧ (1...𝑛) ∈ Fin) → ((♯‘∅) < (♯‘(1...𝑛)) ↔ ∅ ≺ (1...𝑛)))
54	6, 52, 53	mp2an 693	. . . . . . . . . . 11 ⊢ ((♯‘∅) < (♯‘(1...𝑛)) ↔ ∅ ≺ (1...𝑛))
55	51, 54	sylib 218	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ∅ ≺ (1...𝑛))
56	55	anim1i 616	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → (∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴))
57		sdomentr 9042	. . . . . . . . . . 11 ⊢ ((∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → ∅ ≺ 𝐴)
58		sdomnen 8921	. . . . . . . . . . 11 ⊢ (∅ ≺ 𝐴 → ¬ ∅ ≈ 𝐴)
59	57, 58	syl 17	. . . . . . . . . 10 ⊢ ((∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → ¬ ∅ ≈ 𝐴)
60		en0r 8960	. . . . . . . . . . 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 616	. . . . . . . 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 1850	. . . . 5 ⊢ (∃𝑛(¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ ∃𝑛(𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
68		19.42v 1955	. . . . 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 915	. 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 848   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3062  ∅c0 4274   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360   ≈ cen 8883   ≺ csdm 8885  Fincfn 8886  0cc0 11029  1c1 11030   < clt 11170  ℕcn 12165  ℕ₀cn0 12428  ...cfz 13452  ♯chash 14283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-oadd 8402  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-n0 12429  df-xnn0 12502  df-z 12516  df-uz 12780  df-fz 13453  df-hash 14284

This theorem is referenced by: (None)