Theorem ctinfom 12370

Description: A condition for a set being countably infinite. Restates ennnfone 12367 in terms of ω and function image. Like ennnfone 12367 the condition can be summarized as 𝐴 being countable, infinite, and having decidable equality. (Contributed by Jim Kingdon, 7-Aug-2023.)

Assertion

Ref	Expression
ctinfom	⊢ (𝐴 ≈ ℕ ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛))))

Distinct variable groups:   𝐴,𝑓,𝑛   𝑥,𝐴,𝑦   𝑓,𝑘,𝑛

Allowed substitution hint:   𝐴(𝑘)

Proof of Theorem ctinfom

Dummy variables 𝑎 𝑑 𝑖 𝑚 𝑔 𝑏 𝑐 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ennnfone 12367	. . . 4 ⊢ (𝐴 ≈ ℕ ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑔(𝑔:ℕ0–onto→𝐴 ∧ ∀𝑚 ∈ ℕ0 ∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)(𝑔‘𝑗) ≠ (𝑔‘𝑖))))
2	1	simplbi 272	. . 3 ⊢ (𝐴 ≈ ℕ → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
3		nnenom 10377	. . . . . . 7 ⊢ ℕ ≈ ω
4		entr 6758	. . . . . . 7 ⊢ ((𝐴 ≈ ℕ ∧ ℕ ≈ ω) → 𝐴 ≈ ω)
5	3, 4	mpan2 423	. . . . . 6 ⊢ (𝐴 ≈ ℕ → 𝐴 ≈ ω)
6	5	ensymd 6757	. . . . 5 ⊢ (𝐴 ≈ ℕ → ω ≈ 𝐴)
7		bren 6721	. . . . 5 ⊢ (ω ≈ 𝐴 ↔ ∃𝑓 𝑓:ω–1-1-onto→𝐴)
8	6, 7	sylib 121	. . . 4 ⊢ (𝐴 ≈ ℕ → ∃𝑓 𝑓:ω–1-1-onto→𝐴)
9		f1ofo 5447	. . . . . . . 8 ⊢ (𝑓:ω–1-1-onto→𝐴 → 𝑓:ω–onto→𝐴)
10	9	adantl 275	. . . . . . 7 ⊢ ((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) → 𝑓:ω–onto→𝐴)
11		simpr 109	. . . . . . . . 9 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → 𝑛 ∈ ω)
12		nnord 4594	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ω → Ord 𝑛)
13	12	adantl 275	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → Ord 𝑛)
14		ordirr 4524	. . . . . . . . . . 11 ⊢ (Ord 𝑛 → ¬ 𝑛 ∈ 𝑛)
15	13, 14	syl 14	. . . . . . . . . 10 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → ¬ 𝑛 ∈ 𝑛)
16		f1of1 5439	. . . . . . . . . . . 12 ⊢ (𝑓:ω–1-1-onto→𝐴 → 𝑓:ω–1-1→𝐴)
17	16	ad2antlr 486	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → 𝑓:ω–1-1→𝐴)
18		omelon 4591	. . . . . . . . . . . . 13 ⊢ ω ∈ On
19	18	onelssi 4412	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ω → 𝑛 ⊆ ω)
20	19	adantl 275	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω)
21		f1elima 5749	. . . . . . . . . . 11 ⊢ ((𝑓:ω–1-1→𝐴 ∧ 𝑛 ∈ ω ∧ 𝑛 ⊆ ω) → ((𝑓‘𝑛) ∈ (𝑓 “ 𝑛) ↔ 𝑛 ∈ 𝑛))
22	17, 11, 20, 21	syl3anc 1233	. . . . . . . . . 10 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → ((𝑓‘𝑛) ∈ (𝑓 “ 𝑛) ↔ 𝑛 ∈ 𝑛))
23	15, 22	mtbird 668	. . . . . . . . 9 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → ¬ (𝑓‘𝑛) ∈ (𝑓 “ 𝑛))
24		fveq2 5494	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (𝑓‘𝑘) = (𝑓‘𝑛))
25	24	eleq1d 2239	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → ((𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ (𝑓‘𝑛) ∈ (𝑓 “ 𝑛)))
26	25	notbid 662	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ ¬ (𝑓‘𝑛) ∈ (𝑓 “ 𝑛)))
27	26	rspcev 2834	. . . . . . . . 9 ⊢ ((𝑛 ∈ ω ∧ ¬ (𝑓‘𝑛) ∈ (𝑓 “ 𝑛)) → ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛))
28	11, 23, 27	syl2anc 409	. . . . . . . 8 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛))
29	28	ralrimiva 2543	. . . . . . 7 ⊢ ((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛))
30	10, 29	jca 304	. . . . . 6 ⊢ ((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) → (𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛)))
31	30	ex 114	. . . . 5 ⊢ (𝐴 ≈ ℕ → (𝑓:ω–1-1-onto→𝐴 → (𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛))))
32	31	eximdv 1873	. . . 4 ⊢ (𝐴 ≈ ℕ → (∃𝑓 𝑓:ω–1-1-onto→𝐴 → ∃𝑓(𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛))))
33	8, 32	mpd 13	. . 3 ⊢ (𝐴 ≈ ℕ → ∃𝑓(𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛)))
34	2, 33	jca 304	. 2 ⊢ (𝐴 ≈ ℕ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛))))
35		oveq1 5857	. . . . . . . . 9 ⊢ (𝑏 = 𝑎 → (𝑏 + 1) = (𝑎 + 1))
36	35	cbvmptv 4083	. . . . . . . 8 ⊢ (𝑏 ∈ ℤ ↦ (𝑏 + 1)) = (𝑎 ∈ ℤ ↦ (𝑎 + 1))
37		freceq1 6368	. . . . . . . 8 ⊢ ((𝑏 ∈ ℤ ↦ (𝑏 + 1)) = (𝑎 ∈ ℤ ↦ (𝑎 + 1)) → frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) = frec((𝑎 ∈ ℤ ↦ (𝑎 + 1)), 0))
38	36, 37	ax-mp 5	. . . . . . 7 ⊢ frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) = frec((𝑎 ∈ ℤ ↦ (𝑎 + 1)), 0)
39		eqid 2170	. . . . . . 7 ⊢ (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) = (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))
40		simpl 108	. . . . . . 7 ⊢ ((𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛)) → 𝑓:ω–onto→𝐴)
41		fveq2 5494	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑑 → (𝑓‘𝑘) = (𝑓‘𝑑))
42	41	eleq1d 2239	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑑 → ((𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛)))
43	42	notbid 662	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑑 → (¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛)))
44	43	cbvrexv 2697	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ ∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛))
45	44	ralbii 2476	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ ∀𝑛 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛))
46		imaeq2 4947	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑐 → (𝑓 “ 𝑛) = (𝑓 “ 𝑐))
47	46	eleq2d 2240	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑐 → ((𝑓‘𝑑) ∈ (𝑓 “ 𝑛) ↔ (𝑓‘𝑑) ∈ (𝑓 “ 𝑐)))
48	47	notbid 662	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑐 → (¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛) ↔ ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑐)))
49	48	rexbidv 2471	. . . . . . . . . 10 ⊢ (𝑛 = 𝑐 → (∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛) ↔ ∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑐)))
50	49	cbvralv 2696	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛) ↔ ∀𝑐 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑐))
51	45, 50	sylbb 122	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛) → ∀𝑐 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑐))
52	51	adantl 275	. . . . . . 7 ⊢ ((𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛)) → ∀𝑐 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑐))
53	38, 39, 40, 52	ctinfomlemom 12369	. . . . . 6 ⊢ ((𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛)) → ((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)):ℕ0–onto→𝐴 ∧ ∀𝑚 ∈ ℕ0 ∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗) ≠ ((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖)))
54		vex 2733	. . . . . . . 8 ⊢ 𝑓 ∈ V
55		frecex 6370	. . . . . . . . 9 ⊢ frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) ∈ V
56	55	cnvex 5147	. . . . . . . 8 ⊢ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) ∈ V
57	54, 56	coex 5154	. . . . . . 7 ⊢ (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) ∈ V
58		foeq1 5414	. . . . . . . 8 ⊢ (𝑔 = (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔:ℕ0–onto→𝐴 ↔ (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)):ℕ0–onto→𝐴))
59		fveq1 5493	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔‘𝑗) = ((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗))
60		fveq1 5493	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔‘𝑖) = ((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖))
61	59, 60	neeq12d 2360	. . . . . . . . . . 11 ⊢ (𝑔 = (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → ((𝑔‘𝑗) ≠ (𝑔‘𝑖) ↔ ((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗) ≠ ((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖)))
62	61	ralbidv 2470	. . . . . . . . . 10 ⊢ (𝑔 = (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (∀𝑖 ∈ (0...𝑚)(𝑔‘𝑗) ≠ (𝑔‘𝑖) ↔ ∀𝑖 ∈ (0...𝑚)((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗) ≠ ((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖)))
63	62	rexbidv 2471	. . . . . . . . 9 ⊢ (𝑔 = (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)(𝑔‘𝑗) ≠ (𝑔‘𝑖) ↔ ∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗) ≠ ((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖)))
64	63	ralbidv 2470	. . . . . . . 8 ⊢ (𝑔 = (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (∀𝑚 ∈ ℕ0 ∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)(𝑔‘𝑗) ≠ (𝑔‘𝑖) ↔ ∀𝑚 ∈ ℕ0 ∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗) ≠ ((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖)))
65	58, 64	anbi12d 470	. . . . . . 7 ⊢ (𝑔 = (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → ((𝑔:ℕ0–onto→𝐴 ∧ ∀𝑚 ∈ ℕ0 ∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)(𝑔‘𝑗) ≠ (𝑔‘𝑖)) ↔ ((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)):ℕ0–onto→𝐴 ∧ ∀𝑚 ∈ ℕ0 ∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗) ≠ ((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖))))
66	57, 65	spcev 2825	. . . . . 6 ⊢ (((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)):ℕ0–onto→𝐴 ∧ ∀𝑚 ∈ ℕ0 ∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗) ≠ ((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖)) → ∃𝑔(𝑔:ℕ0–onto→𝐴 ∧ ∀𝑚 ∈ ℕ0 ∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)(𝑔‘𝑗) ≠ (𝑔‘𝑖)))
67	53, 66	syl 14	. . . . 5 ⊢ ((𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛)) → ∃𝑔(𝑔:ℕ0–onto→𝐴 ∧ ∀𝑚 ∈ ℕ0 ∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)(𝑔‘𝑗) ≠ (𝑔‘𝑖)))
68	67	exlimiv 1591	. . . 4 ⊢ (∃𝑓(𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛)) → ∃𝑔(𝑔:ℕ0–onto→𝐴 ∧ ∀𝑚 ∈ ℕ0 ∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)(𝑔‘𝑗) ≠ (𝑔‘𝑖)))
69	68	anim2i 340	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛))) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑔(𝑔:ℕ0–onto→𝐴 ∧ ∀𝑚 ∈ ℕ0 ∃𝑗 ∈ ℕ0 ∀𝑖 ∈ (0...𝑚)(𝑔‘𝑗) ≠ (𝑔‘𝑖))))
70	69, 1	sylibr 133	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛))) → 𝐴 ≈ ℕ)
71	34, 70	impbii 125	1 ⊢ (𝐴 ≈ ℕ ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛))))

Syntax hints:  ¬ wn 3   ∧ wa 103   ↔ wb 104  DECID wdc 829   = wceq 1348  ∃wex 1485   ∈ wcel 2141   ≠ wne 2340  ∀wral 2448  ∃wrex 2449   ⊆ wss 3121   class class class wbr 3987   ↦ cmpt 4048  Ord word 4345  ωcom 4572  ^◡ccnv 4608   “ cima 4612   ∘ ccom 4613  –1-1→wf1 5193  –onto→wfo 5194  –1-1-onto→wf1o 5195  ‘cfv 5196  (class class class)co 5850  freccfrec 6366   ≈ cen 6712  0cc0 7761  1c1 7762   + caddc 7764  ℕcn 8865  ℕ₀cn0 9122  ℤcz 9199  ...cfz 9952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-addcom 7861  ax-addass 7863  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-0id 7869  ax-rnegex 7870  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-ltadd 7877

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-er 6509  df-pm 6625  df-en 6715  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-inn 8866  df-n0 9123  df-z 9200  df-uz 9475  df-fz 9953  df-seqfrec 10389

This theorem is referenced by:  ctinf  12372