Theorem ctinfom 13263

Description: A condition for a set being countably infinite. Restates ennnfone 13260 in terms of ω and function image. Like ennnfone 13260 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 13260	. . . 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 10820	. . . . . . 7 ⊢ ℕ ≈ ω
4		entr 7037	. . . . . . 7 ⊢ ((𝐴 ≈ ℕ ∧ ℕ ≈ ω) → 𝐴 ≈ ω)
5	3, 4	mpan2 425	. . . . . 6 ⊢ (𝐴 ≈ ℕ → 𝐴 ≈ ω)
6	5	ensymd 7036	. . . . 5 ⊢ (𝐴 ≈ ℕ → ω ≈ 𝐴)
7		bren 6996	. . . . 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 5626	. . . . . . . 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 4739	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ω → Ord 𝑛)
13	12	adantl 277	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → Ord 𝑛)
14		ordirr 4669	. . . . . . . . . . 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 5618	. . . . . . . . . . . 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 4736	. . . . . . . . . . . . 13 ⊢ ω ∈ On
19	18	onelssi 4555	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ω → 𝑛 ⊆ ω)
20	19	adantl 277	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω)
21		f1elima 5952	. . . . . . . . . . 11 ⊢ ((𝑓:ω–1-1→𝐴 ∧ 𝑛 ∈ ω ∧ 𝑛 ⊆ ω) → ((𝑓‘𝑛) ∈ (𝑓 “ 𝑛) ↔ 𝑛 ∈ 𝑛))
22	17, 11, 20, 21	syl3anc 1274	. . . . . . . . . 10 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → ((𝑓‘𝑛) ∈ (𝑓 “ 𝑛) ↔ 𝑛 ∈ 𝑛))
23	15, 22	mtbird 680	. . . . . . . . 9 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → ¬ (𝑓‘𝑛) ∈ (𝑓 “ 𝑛))
24		fveq2 5675	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (𝑓‘𝑘) = (𝑓‘𝑛))
25	24	eleq1d 2303	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → ((𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ (𝑓‘𝑛) ∈ (𝑓 “ 𝑛)))
26	25	notbid 673	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ ¬ (𝑓‘𝑛) ∈ (𝑓 “ 𝑛)))
27	26	rspcev 2923	. . . . . . . . 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 2617	. . . . . . 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 1929	. . . 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 6065	. . . . . . . . 9 ⊢ (𝑏 = 𝑎 → (𝑏 + 1) = (𝑎 + 1))
36	35	cbvmptv 4211	. . . . . . . 8 ⊢ (𝑏 ∈ ℤ ↦ (𝑏 + 1)) = (𝑎 ∈ ℤ ↦ (𝑎 + 1))
37		freceq1 6636	. . . . . . . 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 2234	. . . . . . 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 5675	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑑 → (𝑓‘𝑘) = (𝑓‘𝑑))
42	41	eleq1d 2303	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑑 → ((𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛)))
43	42	notbid 673	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑑 → (¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛)))
44	43	cbvrexv 2781	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ ∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛))
45	44	ralbii 2550	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ ∀𝑛 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛))
46		imaeq2 5102	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑐 → (𝑓 “ 𝑛) = (𝑓 “ 𝑐))
47	46	eleq2d 2304	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑐 → ((𝑓‘𝑑) ∈ (𝑓 “ 𝑛) ↔ (𝑓‘𝑑) ∈ (𝑓 “ 𝑐)))
48	47	notbid 673	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑐 → (¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛) ↔ ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑐)))
49	48	rexbidv 2545	. . . . . . . . . 10 ⊢ (𝑛 = 𝑐 → (∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛) ↔ ∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑐)))
50	49	cbvralv 2780	. . . . . . . . 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 13262	. . . . . 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 2818	. . . . . . . 8 ⊢ 𝑓 ∈ V
55		frecex 6638	. . . . . . . . 9 ⊢ frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) ∈ V
56	55	cnvex 5306	. . . . . . . 8 ⊢ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) ∈ V
57	54, 56	coex 5313	. . . . . . 7 ⊢ (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) ∈ V
58		foeq1 5591	. . . . . . . 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 5674	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔‘𝑗) = ((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗))
60		fveq1 5674	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔‘𝑖) = ((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖))
61	59, 60	neeq12d 2434	. . . . . . . . . . 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 2544	. . . . . . . . . 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 2545	. . . . . . . . 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 2544	. . . . . . . 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 2914	. . . . . 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 1647	. . . 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 842   = wceq 1398  ∃wex 1541   ∈ wcel 2205   ≠ wne 2414  ∀wral 2522  ∃wrex 2523   ⊆ wss 3214   class class class wbr 4114   ↦ cmpt 4176  Ord word 4488  ωcom 4717  ^◡ccnv 4753   “ cima 4757   ∘ ccom 4758  –1-1→wf1 5354  –onto→wfo 5355  –1-1-onto→wf1o 5356  ‘cfv 5357  (class class class)co 6058  freccfrec 6634   ≈ cen 6986  0cc0 8143  1c1 8144   + caddc 8146  ℕcn 9254  ℕ₀cn0 9513  ℤcz 9594  ...cfz 10361

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-er 6780  df-pm 6898  df-en 6989  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-seqfrec 10834

This theorem is referenced by:  ctinf  13265