Theorem rp-isfinite6 43701

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 894	. . . 4 ⊢ (𝐴 = ∅ ∨ ¬ 𝐴 = ∅)
2	1	biantrur 530	. . 3 ⊢ (𝐴 ∈ Fin ↔ ((𝐴 = ∅ ∨ ¬ 𝐴 = ∅) ∧ 𝐴 ∈ Fin))
3		andir 1010	. . 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 8977	. . . . . 6 ⊢ ∅ ∈ Fin
7		eleq1a 2829	. . . . . 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 43700	. . . . . 6 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴)
12		df-rex 3059	. . . . . 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 623	. . . 4 ⊢ ((¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin) ↔ (¬ 𝐴 = ∅ ∧ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)))
15		df-rex 3059	. . . . 5 ⊢ (∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴 ↔ ∃𝑛(𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
16		en0 8953	. . . . . . . . . . . . . 14 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
17		ensymb 8937	. . . . . . . . . . . . . 14 ⊢ (𝐴 ≈ ∅ ↔ ∅ ≈ 𝐴)
18	16, 17	bitr3i 277	. . . . . . . . . . . . 13 ⊢ (𝐴 = ∅ ↔ ∅ ≈ 𝐴)
19	18	notbii 320	. . . . . . . . . . . 12 ⊢ (¬ 𝐴 = ∅ ↔ ¬ ∅ ≈ 𝐴)
20		elnn0 12401	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ0 ↔ (𝑛 ∈ ℕ ∨ 𝑛 = 0))
21	20	anbi1i 624	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) ↔ ((𝑛 ∈ ℕ ∨ 𝑛 = 0) ∧ (1...𝑛) ≈ 𝐴))
22		andir 1010	. . . . . . . . . . . . 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 628	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ (¬ ∅ ≈ 𝐴 ∧ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))))
25		andi 1009	. . . . . . . . . . 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 1094	. . . . . . . . . . 11 ⊢ ((¬ ∅ ≈ 𝐴 ∧ 𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ↔ (¬ ∅ ≈ 𝐴 ∧ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴)))
28		3anass 1094	. . . . . . . . . . 11 ⊢ ((¬ ∅ ≈ 𝐴 ∧ 𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴) ↔ (¬ ∅ ≈ 𝐴 ∧ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)))
29	27, 28	orbi12i 914	. . . . . . . . . 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 7364	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → (1...𝑛) = (1...0))
33		fz10 13459	. . . . . . . . . . . 12 ⊢ (1...0) = ∅
34	32, 33	eqtrdi 2785	. . . . . . . . . . 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 5120	. . . . . . . . . . . 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 857	. . . . . . . . 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 12174	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 < 𝑛)
46		hash0 14288	. . . . . . . . . . . . 13 ⊢ (♯‘∅) = 0
47	46	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (♯‘∅) = 0)
48		nnnn0 12406	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
49		hashfz1 14267	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ0 → (♯‘(1...𝑛)) = 𝑛)
50	48, 49	syl 17	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (♯‘(1...𝑛)) = 𝑛)
51	45, 47, 50	3brtr4d 5128	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (♯‘∅) < (♯‘(1...𝑛)))
52		fzfi 13893	. . . . . . . . . . . 12 ⊢ (1...𝑛) ∈ Fin
53		hashsdom 14302	. . . . . . . . . . . 12 ⊢ ((∅ ∈ Fin ∧ (1...𝑛) ∈ Fin) → ((♯‘∅) < (♯‘(1...𝑛)) ↔ ∅ ≺ (1...𝑛)))
54	6, 52, 53	mp2an 692	. . . . . . . . . . 11 ⊢ ((♯‘∅) < (♯‘(1...𝑛)) ↔ ∅ ≺ (1...𝑛))
55	51, 54	sylib 218	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ∅ ≺ (1...𝑛))
56	55	anim1i 615	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → (∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴))
57		sdomentr 9037	. . . . . . . . . . 11 ⊢ ((∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → ∅ ≺ 𝐴)
58		sdomnen 8916	. . . . . . . . . . 11 ⊢ (∅ ≺ 𝐴 → ¬ ∅ ≈ 𝐴)
59	57, 58	syl 17	. . . . . . . . . 10 ⊢ ((∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → ¬ ∅ ≈ 𝐴)
60		en0r 8955	. . . . . . . . . . 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 615	. . . . . . . 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 1849	. . . . 5 ⊢ (∃𝑛(¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ ∃𝑛(𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
68		19.42v 1954	. . . . 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 914	. 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 847   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃wrex 3058  ∅c0 4283   class class class wbr 5096  ‘cfv 6490  (class class class)co 7356   ≈ cen 8878   ≺ csdm 8880  Fincfn 8881  0cc0 11024  1c1 11025   < clt 11164  ℕcn 12143  ℕ₀cn0 12399  ...cfz 13421  ♯chash 14251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-n0 12400  df-xnn0 12473  df-z 12487  df-uz 12750  df-fz 13422  df-hash 14252

This theorem is referenced by: (None)