Theorem ctinfom 12298

Description: A condition for a set being countably infinite. Restates ennnfone 12295 in terms of ω and function image. Like ennnfone 12295 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 12295	. . . 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 10359	. . . . . . 7 ⊢ ℕ ≈ ω
4		entr 6741	. . . . . . 7 ⊢ ((𝐴 ≈ ℕ ∧ ℕ ≈ ω) → 𝐴 ≈ ω)
5	3, 4	mpan2 422	. . . . . 6 ⊢ (𝐴 ≈ ℕ → 𝐴 ≈ ω)
6	5	ensymd 6740	. . . . 5 ⊢ (𝐴 ≈ ℕ → ω ≈ 𝐴)
7		bren 6704	. . . . 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 5433	. . . . . . . 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 4583	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ω → Ord 𝑛)
13	12	adantl 275	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → Ord 𝑛)
14		ordirr 4513	. . . . . . . . . . 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 5425	. . . . . . . . . . . 12 ⊢ (𝑓:ω–1-1-onto→𝐴 → 𝑓:ω–1-1→𝐴)
17	16	ad2antlr 481	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → 𝑓:ω–1-1→𝐴)
18		omelon 4580	. . . . . . . . . . . . 13 ⊢ ω ∈ On
19	18	onelssi 4401	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ω → 𝑛 ⊆ ω)
20	19	adantl 275	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω)
21		f1elima 5735	. . . . . . . . . . 11 ⊢ ((𝑓:ω–1-1→𝐴 ∧ 𝑛 ∈ ω ∧ 𝑛 ⊆ ω) → ((𝑓‘𝑛) ∈ (𝑓 “ 𝑛) ↔ 𝑛 ∈ 𝑛))
22	17, 11, 20, 21	syl3anc 1227	. . . . . . . . . 10 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → ((𝑓‘𝑛) ∈ (𝑓 “ 𝑛) ↔ 𝑛 ∈ 𝑛))
23	15, 22	mtbird 663	. . . . . . . . 9 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → ¬ (𝑓‘𝑛) ∈ (𝑓 “ 𝑛))
24		fveq2 5480	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (𝑓‘𝑘) = (𝑓‘𝑛))
25	24	eleq1d 2233	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → ((𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ (𝑓‘𝑛) ∈ (𝑓 “ 𝑛)))
26	25	notbid 657	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ ¬ (𝑓‘𝑛) ∈ (𝑓 “ 𝑛)))
27	26	rspcev 2825	. . . . . . . . 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 2537	. . . . . . 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 1867	. . . 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 5843	. . . . . . . . 9 ⊢ (𝑏 = 𝑎 → (𝑏 + 1) = (𝑎 + 1))
36	35	cbvmptv 4072	. . . . . . . 8 ⊢ (𝑏 ∈ ℤ ↦ (𝑏 + 1)) = (𝑎 ∈ ℤ ↦ (𝑎 + 1))
37		freceq1 6351	. . . . . . . 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 2164	. . . . . . 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 5480	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑑 → (𝑓‘𝑘) = (𝑓‘𝑑))
42	41	eleq1d 2233	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑑 → ((𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛)))
43	42	notbid 657	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑑 → (¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛)))
44	43	cbvrexv 2690	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ ∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛))
45	44	ralbii 2470	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ ∀𝑛 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛))
46		imaeq2 4936	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑐 → (𝑓 “ 𝑛) = (𝑓 “ 𝑐))
47	46	eleq2d 2234	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑐 → ((𝑓‘𝑑) ∈ (𝑓 “ 𝑛) ↔ (𝑓‘𝑑) ∈ (𝑓 “ 𝑐)))
48	47	notbid 657	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑐 → (¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛) ↔ ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑐)))
49	48	rexbidv 2465	. . . . . . . . . 10 ⊢ (𝑛 = 𝑐 → (∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛) ↔ ∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑐)))
50	49	cbvralv 2689	. . . . . . . . 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 12297	. . . . . 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 2724	. . . . . . . 8 ⊢ 𝑓 ∈ V
55		frecex 6353	. . . . . . . . 9 ⊢ frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) ∈ V
56	55	cnvex 5136	. . . . . . . 8 ⊢ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) ∈ V
57	54, 56	coex 5143	. . . . . . 7 ⊢ (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) ∈ V
58		foeq1 5400	. . . . . . . 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 5479	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔‘𝑗) = ((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗))
60		fveq1 5479	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔‘𝑖) = ((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖))
61	59, 60	neeq12d 2354	. . . . . . . . . . 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 2464	. . . . . . . . . 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 2465	. . . . . . . . 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 2464	. . . . . . . 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 465	. . . . . . 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 2816	. . . . . 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 1585	. . . 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 824   = wceq 1342  ∃wex 1479   ∈ wcel 2135   ≠ wne 2334  ∀wral 2442  ∃wrex 2443   ⊆ wss 3111   class class class wbr 3976   ↦ cmpt 4037  Ord word 4334  ωcom 4561  ^◡ccnv 4597   “ cima 4601   ∘ ccom 4602  –1-1→wf1 5179  –onto→wfo 5180  –1-1-onto→wf1o 5181  ‘cfv 5182  (class class class)co 5836  freccfrec 6349   ≈ cen 6695  0cc0 7744  1c1 7745   + caddc 7747  ℕcn 8848  ℕ₀cn0 9105  ℤcz 9182  ...cfz 9935

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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-addcom 7844  ax-addass 7846  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-0id 7852  ax-rnegex 7853  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-ltadd 7860

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-if 3516  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-iord 4338  df-on 4340  df-ilim 4341  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-frec 6350  df-er 6492  df-pm 6608  df-en 6698  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-inn 8849  df-n0 9106  df-z 9183  df-uz 9458  df-fz 9936  df-seqfrec 10371

This theorem is referenced by:  ctinf  12300