Theorem rp-isfinite6 36883

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 Richard Penner, 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 430	. . . 4 ⊢ (𝐴 = ∅ ∨ ¬ 𝐴 = ∅)
2	1	biantrur 526	. . 3 ⊢ (𝐴 ∈ Fin ↔ ((𝐴 = ∅ ∨ ¬ 𝐴 = ∅) ∧ 𝐴 ∈ Fin))
3		andir 908	. . 3 ⊢ (((𝐴 = ∅ ∨ ¬ 𝐴 = ∅) ∧ 𝐴 ∈ Fin) ↔ ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ∨ (¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin)))
4	2, 3	bitri 263	. 2 ⊢ (𝐴 ∈ Fin ↔ ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ∨ (¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin)))
5		simpl 472	. . . 4 ⊢ ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) → 𝐴 = ∅)
6		0fin 8073	. . . . . 6 ⊢ ∅ ∈ Fin
7		eleq1a 2683	. . . . . 6 ⊢ (∅ ∈ Fin → (𝐴 = ∅ → 𝐴 ∈ Fin))
8	6, 7	ax-mp 5	. . . . 5 ⊢ (𝐴 = ∅ → 𝐴 ∈ Fin)
9	8	ancli 572	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 = ∅ ∧ 𝐴 ∈ Fin))
10	5, 9	impbii 198	. . 3 ⊢ ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ↔ 𝐴 = ∅)
11		rp-isfinite5 36882	. . . . . 6 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴)
12		df-rex 2902	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴 ↔ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴))
13	11, 12	bitri 263	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴))
14	13	anbi2i 726	. . . 4 ⊢ ((¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin) ↔ (¬ 𝐴 = ∅ ∧ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)))
15		df-rex 2902	. . . . 5 ⊢ (∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴 ↔ ∃𝑛(𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
16		en0 7905	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
17	16	bicomi 213	. . . . . . . . . . . . . 14 ⊢ (𝐴 = ∅ ↔ 𝐴 ≈ ∅)
18		ensymb 7890	. . . . . . . . . . . . . 14 ⊢ (𝐴 ≈ ∅ ↔ ∅ ≈ 𝐴)
19	17, 18	bitri 263	. . . . . . . . . . . . 13 ⊢ (𝐴 = ∅ ↔ ∅ ≈ 𝐴)
20	19	notbii 309	. . . . . . . . . . . 12 ⊢ (¬ 𝐴 = ∅ ↔ ¬ ∅ ≈ 𝐴)
21		elnn0 11171	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ0 ↔ (𝑛 ∈ ℕ ∨ 𝑛 = 0))
22	21	anbi1i 727	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) ↔ ((𝑛 ∈ ℕ ∨ 𝑛 = 0) ∧ (1...𝑛) ≈ 𝐴))
23		andir 908	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ ∨ 𝑛 = 0) ∧ (1...𝑛) ≈ 𝐴) ↔ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)))
24	22, 23	bitri 263	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) ↔ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)))
25	20, 24	anbi12i 729	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ (¬ ∅ ≈ 𝐴 ∧ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))))
26		andi 907	. . . . . . . . . . 11 ⊢ ((¬ ∅ ≈ 𝐴 ∧ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))) ↔ ((¬ ∅ ≈ 𝐴 ∧ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴)) ∨ (¬ ∅ ≈ 𝐴 ∧ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))))
27	25, 26	bitri 263	. . . . . . . . . 10 ⊢ ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ ((¬ ∅ ≈ 𝐴 ∧ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴)) ∨ (¬ ∅ ≈ 𝐴 ∧ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))))
28		3anass 1035	. . . . . . . . . . 11 ⊢ ((¬ ∅ ≈ 𝐴 ∧ 𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ↔ (¬ ∅ ≈ 𝐴 ∧ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴)))
29		3anass 1035	. . . . . . . . . . 11 ⊢ ((¬ ∅ ≈ 𝐴 ∧ 𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴) ↔ (¬ ∅ ≈ 𝐴 ∧ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)))
30	28, 29	orbi12i 542	. . . . . . . . . 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))))
31	27, 30	sylbb2 227	. . . . . . . . 9 ⊢ ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) → ((¬ ∅ ≈ 𝐴 ∧ 𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (¬ ∅ ≈ 𝐴 ∧ 𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)))
32		simp2 1055	. . . . . . . . . 10 ⊢ ((¬ ∅ ≈ 𝐴 ∧ 𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → 𝑛 ∈ ℕ)
33		oveq2 6557	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → (1...𝑛) = (1...0))
34		fz10 12233	. . . . . . . . . . . 12 ⊢ (1...0) = ∅
35	33, 34	syl6eq 2660	. . . . . . . . . . 11 ⊢ (𝑛 = 0 → (1...𝑛) = ∅)
36		simp2 1055	. . . . . . . . . . . . 13 ⊢ ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → (1...𝑛) = ∅)
37		simp3 1056	. . . . . . . . . . . . 13 ⊢ ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → (1...𝑛) ≈ 𝐴)
38	36, 37	eqbrtrrd 4607	. . . . . . . . . . . 12 ⊢ ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → ∅ ≈ 𝐴)
39		simp1 1054	. . . . . . . . . . . 12 ⊢ ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → ¬ ∅ ≈ 𝐴)
40	38, 39	pm2.21dd 185	. . . . . . . . . . 11 ⊢ ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → 𝑛 ∈ ℕ)
41	35, 40	syl3an2 1352	. . . . . . . . . 10 ⊢ ((¬ ∅ ≈ 𝐴 ∧ 𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴) → 𝑛 ∈ ℕ)
42	32, 41	jaoi 393	. . . . . . . . 9 ⊢ (((¬ ∅ ≈ 𝐴 ∧ 𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (¬ ∅ ≈ 𝐴 ∧ 𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)) → 𝑛 ∈ ℕ)
43	31, 42	syl 17	. . . . . . . 8 ⊢ ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) → 𝑛 ∈ ℕ)
44		simprr 792	. . . . . . . 8 ⊢ ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) → (1...𝑛) ≈ 𝐴)
45	43, 44	jca 553	. . . . . . 7 ⊢ ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) → (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
46		nngt0 10926	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 < 𝑛)
47		hash0 13019	. . . . . . . . . . . . 13 ⊢ (#‘∅) = 0
48	47	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (#‘∅) = 0)
49		nnnn0 11176	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
50		hashfz1 12996	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ0 → (#‘(1...𝑛)) = 𝑛)
51	49, 50	syl 17	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (#‘(1...𝑛)) = 𝑛)
52	46, 48, 51	3brtr4d 4615	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (#‘∅) < (#‘(1...𝑛)))
53		fzfi 12633	. . . . . . . . . . . 12 ⊢ (1...𝑛) ∈ Fin
54		hashsdom 13031	. . . . . . . . . . . 12 ⊢ ((∅ ∈ Fin ∧ (1...𝑛) ∈ Fin) → ((#‘∅) < (#‘(1...𝑛)) ↔ ∅ ≺ (1...𝑛)))
55	6, 53, 54	mp2an 704	. . . . . . . . . . 11 ⊢ ((#‘∅) < (#‘(1...𝑛)) ↔ ∅ ≺ (1...𝑛))
56	52, 55	sylib 207	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ∅ ≺ (1...𝑛))
57	56	anim1i 590	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → (∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴))
58		sdomentr 7979	. . . . . . . . . . 11 ⊢ ((∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → ∅ ≺ 𝐴)
59		sdomnen 7870	. . . . . . . . . . 11 ⊢ (∅ ≺ 𝐴 → ¬ ∅ ≈ 𝐴)
60	58, 59	syl 17	. . . . . . . . . 10 ⊢ ((∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → ¬ ∅ ≈ 𝐴)
61		ensymb 7890	. . . . . . . . . . . 12 ⊢ (∅ ≈ 𝐴 ↔ 𝐴 ≈ ∅)
62	61, 16	bitri 263	. . . . . . . . . . 11 ⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅)
63	62	notbii 309	. . . . . . . . . 10 ⊢ (¬ ∅ ≈ 𝐴 ↔ ¬ 𝐴 = ∅)
64	60, 63	sylib 207	. . . . . . . . 9 ⊢ ((∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → ¬ 𝐴 = ∅)
65	57, 64	syl 17	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → ¬ 𝐴 = ∅)
66	49	anim1i 590	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴))
67	65, 66	jca 553	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → (¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)))
68	45, 67	impbii 198	. . . . . 6 ⊢ ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
69	68	exbii 1764	. . . . 5 ⊢ (∃𝑛(¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ ∃𝑛(𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
70		19.42v 1905	. . . . 5 ⊢ (∃𝑛(¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ (¬ 𝐴 = ∅ ∧ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)))
71	15, 69, 70	3bitr2ri 288	. . . 4 ⊢ ((¬ 𝐴 = ∅ ∧ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴)
72	14, 71	bitri 263	. . 3 ⊢ ((¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin) ↔ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴)
73	10, 72	orbi12i 542	. 2 ⊢ (((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ∨ (¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin)) ↔ (𝐴 = ∅ ∨ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴))
74	4, 73	bitri 263	1 ⊢ (𝐴 ∈ Fin ↔ (𝐴 = ∅ ∨ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  ∅c0 3874   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549   ≈ cen 7838   ≺ csdm 7840  Fincfn 7841  0cc0 9815  1c1 9816   < clt 9953  ℕcn 10897  ℕ₀cn0 11169  ...cfz 12197  #chash 12979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980

This theorem is referenced by: (None)