Theorem ctinfom 12447

Description: A condition for a set being countably infinite. Restates ennnfone 12444 in terms of ω and function image. Like ennnfone 12444 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 12444	. . . 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 10452	. . . . . . 7 ⊢ ℕ ≈ ω
4		entr 6802	. . . . . . 7 ⊢ ((𝐴 ≈ ℕ ∧ ℕ ≈ ω) → 𝐴 ≈ ω)
5	3, 4	mpan2 425	. . . . . 6 ⊢ (𝐴 ≈ ℕ → 𝐴 ≈ ω)
6	5	ensymd 6801	. . . . 5 ⊢ (𝐴 ≈ ℕ → ω ≈ 𝐴)
7		bren 6765	. . . . 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 5483	. . . . . . . 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 4626	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ω → Ord 𝑛)
13	12	adantl 277	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → Ord 𝑛)
14		ordirr 4556	. . . . . . . . . . 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 5475	. . . . . . . . . . . 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 4623	. . . . . . . . . . . . 13 ⊢ ω ∈ On
19	18	onelssi 4444	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ω → 𝑛 ⊆ ω)
20	19	adantl 277	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω)
21		f1elima 5790	. . . . . . . . . . 11 ⊢ ((𝑓:ω–1-1→𝐴 ∧ 𝑛 ∈ ω ∧ 𝑛 ⊆ ω) → ((𝑓‘𝑛) ∈ (𝑓 “ 𝑛) ↔ 𝑛 ∈ 𝑛))
22	17, 11, 20, 21	syl3anc 1249	. . . . . . . . . 10 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → ((𝑓‘𝑛) ∈ (𝑓 “ 𝑛) ↔ 𝑛 ∈ 𝑛))
23	15, 22	mtbird 674	. . . . . . . . 9 ⊢ (((𝐴 ≈ ℕ ∧ 𝑓:ω–1-1-onto→𝐴) ∧ 𝑛 ∈ ω) → ¬ (𝑓‘𝑛) ∈ (𝑓 “ 𝑛))
24		fveq2 5530	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (𝑓‘𝑘) = (𝑓‘𝑛))
25	24	eleq1d 2258	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → ((𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ (𝑓‘𝑛) ∈ (𝑓 “ 𝑛)))
26	25	notbid 668	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ ¬ (𝑓‘𝑛) ∈ (𝑓 “ 𝑛)))
27	26	rspcev 2856	. . . . . . . . 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 2563	. . . . . . 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 1891	. . . 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 5898	. . . . . . . . 9 ⊢ (𝑏 = 𝑎 → (𝑏 + 1) = (𝑎 + 1))
36	35	cbvmptv 4114	. . . . . . . 8 ⊢ (𝑏 ∈ ℤ ↦ (𝑏 + 1)) = (𝑎 ∈ ℤ ↦ (𝑎 + 1))
37		freceq1 6411	. . . . . . . 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 2189	. . . . . . 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 5530	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑑 → (𝑓‘𝑘) = (𝑓‘𝑑))
42	41	eleq1d 2258	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑑 → ((𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛)))
43	42	notbid 668	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑑 → (¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛)))
44	43	cbvrexv 2719	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ ∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛))
45	44	ralbii 2496	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛) ↔ ∀𝑛 ∈ ω ∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛))
46		imaeq2 4981	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑐 → (𝑓 “ 𝑛) = (𝑓 “ 𝑐))
47	46	eleq2d 2259	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑐 → ((𝑓‘𝑑) ∈ (𝑓 “ 𝑛) ↔ (𝑓‘𝑑) ∈ (𝑓 “ 𝑐)))
48	47	notbid 668	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑐 → (¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛) ↔ ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑐)))
49	48	rexbidv 2491	. . . . . . . . . 10 ⊢ (𝑛 = 𝑐 → (∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑛) ↔ ∃𝑑 ∈ ω ¬ (𝑓‘𝑑) ∈ (𝑓 “ 𝑐)))
50	49	cbvralv 2718	. . . . . . . . 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 12446	. . . . . 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 2755	. . . . . . . 8 ⊢ 𝑓 ∈ V
55		frecex 6413	. . . . . . . . 9 ⊢ frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) ∈ V
56	55	cnvex 5182	. . . . . . . 8 ⊢ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0) ∈ V
57	54, 56	coex 5189	. . . . . . 7 ⊢ (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) ∈ V
58		foeq1 5449	. . . . . . . 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 5529	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔‘𝑗) = ((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑗))
60		fveq1 5529	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0)) → (𝑔‘𝑖) = ((𝑓 ∘ ◡frec((𝑏 ∈ ℤ ↦ (𝑏 + 1)), 0))‘𝑖))
61	59, 60	neeq12d 2380	. . . . . . . . . . 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 2490	. . . . . . . . . 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 2491	. . . . . . . . 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 2490	. . . . . . . 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 2847	. . . . . 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 1609	. . . 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 835   = wceq 1364  ∃wex 1503   ∈ wcel 2160   ≠ wne 2360  ∀wral 2468  ∃wrex 2469   ⊆ wss 3144   class class class wbr 4018   ↦ cmpt 4079  Ord word 4377  ωcom 4604  ^◡ccnv 4640   “ cima 4644   ∘ ccom 4645  –1-1→wf1 5228  –onto→wfo 5229  –1-1-onto→wf1o 5230  ‘cfv 5231  (class class class)co 5891  freccfrec 6409   ≈ cen 6756  0cc0 7829  1c1 7830   + caddc 7832  ℕcn 8937  ℕ₀cn0 9194  ℤcz 9271  ...cfz 10026

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-addcom 7929  ax-addass 7931  ax-distr 7933  ax-i2m1 7934  ax-0lt1 7935  ax-0id 7937  ax-rnegex 7938  ax-cnre 7940  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943  ax-pre-ltadd 7945

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-frec 6410  df-er 6553  df-pm 6669  df-en 6759  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-sub 8148  df-neg 8149  df-inn 8938  df-n0 9195  df-z 9272  df-uz 9547  df-fz 10027  df-seqfrec 10464

This theorem is referenced by:  ctinf  12449