Theorem ctinfom 12429

Description: A condition for a set being countably infinite. Restates ennnfone 12426 in terms of ω and function image. Like ennnfone 12426 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 12426	. . . 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 274	. . 3 ⊢ (𝐴 ≈ ℕ → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
3		nnenom 10434	. . . . . . 7 ⊢ ℕ ≈ ω
4		entr 6784	. . . . . . 7 ⊢ ((𝐴 ≈ ℕ ∧ ℕ ≈ ω) → 𝐴 ≈ ω)
5	3, 4	mpan2 425	. . . . . 6 ⊢ (𝐴 ≈ ℕ → 𝐴 ≈ ω)
6	5	ensymd 6783	. . . . 5 ⊢ (𝐴 ≈ ℕ → ω ≈ 𝐴)
7		bren 6747	. . . . 5 ⊢ (ω ≈ 𝐴 ↔ ∃𝑓 𝑓:ω–1-1-onto→𝐴)
8	6, 7	sylib 122	. . . 4 ⊢ (𝐴 ≈ ℕ → ∃𝑓 𝑓:ω–1-1-onto→𝐴)
9		f1ofo 5469	. . . . . . . 8 ⊢ (𝑓:ω–1-1-onto→𝐴 → 𝑓:ω–onto→𝐴)
10	9	adantl 277	. . . . . . 7 ⊢ ((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) → 𝑓:ω–onto→𝐴)
11		simpr 110	. . . . . . . . 9 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → 𝑛 ∈ ω)
12		nnord 4612	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ω → Ord 𝑛)
13	12	adantl 277	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → Ord 𝑛)
14		ordirr 4542	. . . . . . . . . . 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 5461	. . . . . . . . . . . 12 ⊢ (𝑓:ω–1-1-onto→𝐴 → 𝑓:ω–1-1→𝐴)
17	16	ad2antlr 489	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → 𝑓:ω–1-1→𝐴)
18		omelon 4609	. . . . . . . . . . . . 13 ⊢ ω ∈ On
19	18	onelssi 4430	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ω → 𝑛 ⊆ ω)
20	19	adantl 277	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω)
21		f1elima 5774	. . . . . . . . . . 11 ⊢ ((𝑓:ω–1-1→𝐴 ∧ 𝑛 ∈ ω ∧ 𝑛 ⊆ ω) → ((𝑓‘𝑛) ∈ (𝑓 “ 𝑛) ↔ 𝑛 ∈ 𝑛))
22	17, 11, 20, 21	syl3anc 1238	. . . . . . . . . 10 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → ((𝑓‘𝑛) ∈ (𝑓 “ 𝑛) ↔ 𝑛 ∈ 𝑛))
23	15, 22	mtbird 673	. . . . . . . . 9 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → ¬ (𝑓‘𝑛) ∈ (𝑓 “ 𝑛))
24		fveq2 5516	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (𝑓‘𝑘) = (𝑓‘𝑛))
25	24	eleq1d 2246	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → ((𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ (𝑓‘𝑛) ∈ (𝑓 “ 𝑛)))
26	25	notbid 667	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ ¬ (𝑓‘𝑛) ∈ (𝑓 “ 𝑛)))
27	26	rspcev 2842	. . . . . . . . 9 ⊢ ((𝑛 ∈ ω ∧ ¬ (𝑓‘𝑛) ∈ (𝑓 “ 𝑛)) → ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛))
28	11, 23, 27	syl2anc 411	. . . . . . . 8 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛))
29	28	ralrimiva 2550	. . . . . . 7 ⊢ ((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛))
30	10, 29	jca 306	. . . . . 6 ⊢ ((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) → (𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛)))
31	30	ex 115	. . . . 5 ⊢ (𝐴 ≈ ℕ → (𝑓:ω–1-1-onto→𝐴 → (𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛))))
32	31	eximdv 1880	. . . 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 306	. 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 5882	. . . . . . . . 9 ⊢ (𝑏 = 𝑎 → (𝑏 + 1) = (𝑎 + 1))
36	35	cbvmptv 4100	. . . . . . . 8 ⊢ (𝑏 ∈ ℤ ↦ (𝑏 + 1)) = (𝑎 ∈ ℤ ↦ (𝑎 + 1))
37		freceq1 6393	. . . . . . . 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 2177	. . . . . . 7 ⊢ (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) = (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))
40		simpl 109	. . . . . . 7 ⊢ ((𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛)) → 𝑓:ω–onto→𝐴)
41		fveq2 5516	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑑 → (𝑓‘𝑘) = (𝑓‘𝑑))
42	41	eleq1d 2246	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑑 → ((𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛)))
43	42	notbid 667	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑑 → (¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛)))
44	43	cbvrexv 2705	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ ∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛))
45	44	ralbii 2483	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ ∀𝑛 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛))
46		imaeq2 4967	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑐 → (𝑓 “ 𝑛) = (𝑓 “ 𝑐))
47	46	eleq2d 2247	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑐 → ((𝑓‘𝑑) ∈ (𝑓 “ 𝑛) ↔ (𝑓‘𝑑) ∈ (𝑓 “ 𝑐)))
48	47	notbid 667	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑐 → (¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛) ↔ ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑐)))
49	48	rexbidv 2478	. . . . . . . . . 10 ⊢ (𝑛 = 𝑐 → (∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛) ↔ ∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑐)))
50	49	cbvralv 2704	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛) ↔ ∀𝑐 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑐))
51	45, 50	sylbb 123	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛) → ∀𝑐 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑐))
52	51	adantl 277	. . . . . . 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 12428	. . . . . 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 2741	. . . . . . . 8 ⊢ 𝑓 ∈ V
55		frecex 6395	. . . . . . . . 9 ⊢ frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) ∈ V
56	55	cnvex 5168	. . . . . . . 8 ⊢ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) ∈ V
57	54, 56	coex 5175	. . . . . . 7 ⊢ (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) ∈ V
58		foeq1 5435	. . . . . . . 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 5515	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔‘𝑗) = ((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗))
60		fveq1 5515	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔‘𝑖) = ((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖))
61	59, 60	neeq12d 2367	. . . . . . . . . . 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 2477	. . . . . . . . . 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 2478	. . . . . . . . 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 2477	. . . . . . . 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 473	. . . . . . 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 2833	. . . . . 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 1598	. . . 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 342	. . 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 134	. 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 126	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 104   ↔ wb 105  DECID wdc 834   = wceq 1353  ∃wex 1492   ∈ wcel 2148   ≠ wne 2347  ∀wral 2455  ∃wrex 2456   ⊆ wss 3130   class class class wbr 4004   ↦ cmpt 4065  Ord word 4363  ωcom 4590  ^◡ccnv 4626   “ cima 4630   ∘ ccom 4631  –1-1→wf1 5214  –onto→wfo 5215  –1-1-onto→wf1o 5216  ‘cfv 5217  (class class class)co 5875  freccfrec 6391   ≈ cen 6738  0cc0 7811  1c1 7812   + caddc 7814  ℕcn 8919  ℕ₀cn0 9176  ℤcz 9253  ...cfz 10008

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-ltadd 7927

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-er 6535  df-pm 6651  df-en 6741  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-inn 8920  df-n0 9177  df-z 9254  df-uz 9529  df-fz 10009  df-seqfrec 10446

This theorem is referenced by:  ctinf  12431