Theorem rp-isfinite6 41780

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 531	. . 3 ⊢ (𝐴 ∈ Fin ↔ ((𝐴 = ∅ ∨ ¬ 𝐴 = ∅) ∧ 𝐴 ∈ Fin))
3		andir 1007	. . 3 ⊢ (((𝐴 = ∅ ∨ ¬ 𝐴 = ∅) ∧ 𝐴 ∈ Fin) ↔ ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ∨ (¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin)))
4	2, 3	bitri 274	. 2 ⊢ (𝐴 ∈ Fin ↔ ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ∨ (¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin)))
5		simpl 483	. . . 4 ⊢ ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) → 𝐴 = ∅)
6		0fin 9115	. . . . . 6 ⊢ ∅ ∈ Fin
7		eleq1a 2833	. . . . . 6 ⊢ (∅ ∈ Fin → (𝐴 = ∅ → 𝐴 ∈ Fin))
8	6, 7	ax-mp 5	. . . . 5 ⊢ (𝐴 = ∅ → 𝐴 ∈ Fin)
9	8	ancli 549	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 = ∅ ∧ 𝐴 ∈ Fin))
10	5, 9	impbii 208	. . 3 ⊢ ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ↔ 𝐴 = ∅)
11		rp-isfinite5 41779	. . . . . 6 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴)
12		df-rex 3074	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴 ↔ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴))
13	11, 12	bitri 274	. . . . 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 3074	. . . . 5 ⊢ (∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴 ↔ ∃𝑛(𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
16		en0 8957	. . . . . . . . . . . . . 14 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
17		ensymb 8942	. . . . . . . . . . . . . 14 ⊢ (𝐴 ≈ ∅ ↔ ∅ ≈ 𝐴)
18	16, 17	bitr3i 276	. . . . . . . . . . . . 13 ⊢ (𝐴 = ∅ ↔ ∅ ≈ 𝐴)
19	18	notbii 319	. . . . . . . . . . . 12 ⊢ (¬ 𝐴 = ∅ ↔ ¬ ∅ ≈ 𝐴)
20		elnn0 12415	. . . . . . . . . . . . . 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 1007	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ ∨ 𝑛 = 0) ∧ (1...𝑛) ≈ 𝐴) ↔ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)))
23	21, 22	bitri 274	. . . . . . . . . . . 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 1006	. . . . . . . . . . 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 274	. . . . . . . . . 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 237	. . . . . . . . 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 7365	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → (1...𝑛) = (1...0))
33		fz10 13462	. . . . . . . . . . . 12 ⊢ (1...0) = ∅
34	32, 33	eqtrdi 2792	. . . . . . . . . . 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 5129	. . . . . . . . . . . 12 ⊢ ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → ∅ ≈ 𝐴)
38		simp1 1136	. . . . . . . . . . . 12 ⊢ ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → ¬ ∅ ≈ 𝐴)
39	37, 38	pm2.21dd 194	. . . . . . . . . . 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 855	. . . . . . . . 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 771	. . . . . . . 8 ⊢ ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) → (1...𝑛) ≈ 𝐴)
44	42, 43	jca 512	. . . . . . 7 ⊢ ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) → (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
45		nngt0 12184	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 < 𝑛)
46		hash0 14267	. . . . . . . . . . . . 13 ⊢ (♯‘∅) = 0
47	46	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (♯‘∅) = 0)
48		nnnn0 12420	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
49		hashfz1 14246	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ0 → (♯‘(1...𝑛)) = 𝑛)
50	48, 49	syl 17	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (♯‘(1...𝑛)) = 𝑛)
51	45, 47, 50	3brtr4d 5137	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (♯‘∅) < (♯‘(1...𝑛)))
52		fzfi 13877	. . . . . . . . . . . 12 ⊢ (1...𝑛) ∈ Fin
53		hashsdom 14281	. . . . . . . . . . . 12 ⊢ ((∅ ∈ Fin ∧ (1...𝑛) ∈ Fin) → ((♯‘∅) < (♯‘(1...𝑛)) ↔ ∅ ≺ (1...𝑛)))
54	6, 52, 53	mp2an 690	. . . . . . . . . . 11 ⊢ ((♯‘∅) < (♯‘(1...𝑛)) ↔ ∅ ≺ (1...𝑛))
55	51, 54	sylib 217	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ∅ ≺ (1...𝑛))
56	55	anim1i 615	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → (∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴))
57		sdomentr 9055	. . . . . . . . . . 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 319	. . . . . . . . . 10 ⊢ (¬ ∅ ≈ 𝐴 ↔ ¬ 𝐴 = ∅)
62	59, 61	sylib 217	. . . . . . . . 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 512	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → (¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)))
66	44, 65	impbii 208	. . . . . 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 1957	. . . . 5 ⊢ (∃𝑛(¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ (¬ 𝐴 = ∅ ∧ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)))
69	15, 67, 68	3bitr2ri 299	. . . 4 ⊢ ((¬ 𝐴 = ∅ ∧ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴)
70	14, 69	bitri 274	. . 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 274	1 ⊢ (𝐴 ∈ Fin ↔ (𝐴 = ∅ ∨ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∃wrex 3073  ∅c0 4282   class class class wbr 5105  ‘cfv 6496  (class class class)co 7357   ≈ cen 8880   ≺ csdm 8882  Fincfn 8883  0cc0 11051  1c1 11052   < clt 11189  ℕcn 12153  ℕ₀cn0 12413  ...cfz 13424  ♯chash 14230

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-hash 14231

This theorem is referenced by: (None)