Theorem ctinf 12396

Description: A set is countably infinite if and only if it has decidable equality, is countable, and is infinite. (Contributed by Jim Kingdon, 7-Aug-2023.)

Assertion

Ref	Expression
ctinf	⊢ (𝐴 ≈ ℕ ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴))

Distinct variable group:   𝐴,𝑓,𝑦,𝑥

Proof of Theorem ctinf

Dummy variables 𝑎 𝑏 𝑛 𝑘 𝑢 𝑔 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ctinfom 12394	. . . 4 ⊢ (𝐴 ≈ ℕ ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛))))
2	1	simplbi 274	. . 3 ⊢ (𝐴 ≈ ℕ → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
3	1	simprbi 275	. . . 4 ⊢ (𝐴 ≈ ℕ → ∃𝑓(𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛)))
4		simpl 109	. . . . . 6 ⊢ ((𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛)) → 𝑓:ω–onto→𝐴)
5	4	a1i 9	. . . . 5 ⊢ (𝐴 ≈ ℕ → ((𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛)) → 𝑓:ω–onto→𝐴))
6	5	eximdv 1878	. . . 4 ⊢ (𝐴 ≈ ℕ → (∃𝑓(𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛)) → ∃𝑓 𝑓:ω–onto→𝐴))
7	3, 6	mpd 13	. . 3 ⊢ (𝐴 ≈ ℕ → ∃𝑓 𝑓:ω–onto→𝐴)
8		nnenom 10402	. . . . . 6 ⊢ ℕ ≈ ω
9		entr 6774	. . . . . 6 ⊢ ((𝐴 ≈ ℕ ∧ ℕ ≈ ω) → 𝐴 ≈ ω)
10	8, 9	mpan2 425	. . . . 5 ⊢ (𝐴 ≈ ℕ → 𝐴 ≈ ω)
11	10	ensymd 6773	. . . 4 ⊢ (𝐴 ≈ ℕ → ω ≈ 𝐴)
12		endom 6753	. . . 4 ⊢ (ω ≈ 𝐴 → ω ≼ 𝐴)
13	11, 12	syl 14	. . 3 ⊢ (𝐴 ≈ ℕ → ω ≼ 𝐴)
14	2, 7, 13	3jca 1177	. 2 ⊢ (𝐴 ≈ ℕ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴))
15		simp1 997	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
16		3simpb 995	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴))
17		simp2 998	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) → ∃𝑓 𝑓:ω–onto→𝐴)
18		simp2 998	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) → 𝑓:ω–onto→𝐴)
19		simpl1 1000	. . . . . . . . . . . 12 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
20		equequ1 1710	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑢 → (𝑥 = 𝑦 ↔ 𝑢 = 𝑦))
21	20	dcbid 838	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑢 = 𝑦))
22	21	ralbidv 2475	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → (∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ↔ ∀𝑦 ∈ 𝐴 DECID 𝑢 = 𝑦))
23	22	cbvralv 2701	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ↔ ∀𝑢 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑢 = 𝑦)
24	19, 23	sylib 122	. . . . . . . . . . 11 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → ∀𝑢 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑢 = 𝑦)
25		simpl3 1002	. . . . . . . . . . 11 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → ω ≼ 𝐴)
26		fof 5430	. . . . . . . . . . . . . 14 ⊢ (𝑓:ω–onto→𝐴 → 𝑓:ω⟶𝐴)
27		imassrn 4974	. . . . . . . . . . . . . . 15 ⊢ (𝑓 “ 𝑛) ⊆ ran 𝑓
28		frn 5366	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ω⟶𝐴 → ran 𝑓 ⊆ 𝐴)
29	27, 28	sstrid 3164	. . . . . . . . . . . . . 14 ⊢ (𝑓:ω⟶𝐴 → (𝑓 “ 𝑛) ⊆ 𝐴)
30	26, 29	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑓:ω–onto→𝐴 → (𝑓 “ 𝑛) ⊆ 𝐴)
31	30	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → (𝑓 “ 𝑛) ⊆ 𝐴)
32	31	3adantl1 1153	. . . . . . . . . . 11 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → (𝑓 “ 𝑛) ⊆ 𝐴)
33		simpl2 1001	. . . . . . . . . . . . 13 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → 𝑓:ω–onto→𝐴)
34		equequ1 1710	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → (𝑥 = 𝑦 ↔ 𝑎 = 𝑦))
35	34	dcbid 838	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑎 = 𝑦))
36		equequ2 1711	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑏 → (𝑎 = 𝑦 ↔ 𝑎 = 𝑏))
37	36	dcbid 838	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑏 → (DECID 𝑎 = 𝑦 ↔ DECID 𝑎 = 𝑏))
38	35, 37	cbvral2v 2714	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 DECID 𝑎 = 𝑏)
39		ssralv 3217	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 “ 𝑛) ⊆ 𝐴 → (∀𝑏 ∈ 𝐴 DECID 𝑎 = 𝑏 → ∀𝑏 ∈ (𝑓 “ 𝑛)DECID 𝑎 = 𝑏))
40	30, 39	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:ω–onto→𝐴 → (∀𝑏 ∈ 𝐴 DECID 𝑎 = 𝑏 → ∀𝑏 ∈ (𝑓 “ 𝑛)DECID 𝑎 = 𝑏))
41	40	ralimdv 2543	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ω–onto→𝐴 → (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 DECID 𝑎 = 𝑏 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ (𝑓 “ 𝑛)DECID 𝑎 = 𝑏))
42		ssralv 3217	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 “ 𝑛) ⊆ 𝐴 → (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ (𝑓 “ 𝑛)DECID 𝑎 = 𝑏 → ∀𝑎 ∈ (𝑓 “ 𝑛)∀𝑏 ∈ (𝑓 “ 𝑛)DECID 𝑎 = 𝑏))
43	30, 41, 42	sylsyld 58	. . . . . . . . . . . . . 14 ⊢ (𝑓:ω–onto→𝐴 → (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 DECID 𝑎 = 𝑏 → ∀𝑎 ∈ (𝑓 “ 𝑛)∀𝑏 ∈ (𝑓 “ 𝑛)DECID 𝑎 = 𝑏))
44	38, 43	biimtrid 152	. . . . . . . . . . . . 13 ⊢ (𝑓:ω–onto→𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → ∀𝑎 ∈ (𝑓 “ 𝑛)∀𝑏 ∈ (𝑓 “ 𝑛)DECID 𝑎 = 𝑏))
45	33, 19, 44	sylc 62	. . . . . . . . . . . 12 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → ∀𝑎 ∈ (𝑓 “ 𝑛)∀𝑏 ∈ (𝑓 “ 𝑛)DECID 𝑎 = 𝑏)
46		simpr 110	. . . . . . . . . . . . . 14 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → 𝑛 ∈ ω)
47		fofun 5431	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:ω–onto→𝐴 → Fun 𝑓)
48	47	ad2antrr 488	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → Fun 𝑓)
49		ordom 4600	. . . . . . . . . . . . . . . . . . 19 ⊢ Ord ω
50		ordtr 4372	. . . . . . . . . . . . . . . . . . 19 ⊢ (Ord ω → Tr ω)
51	49, 50	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ Tr ω
52		trss 4105	. . . . . . . . . . . . . . . . . 18 ⊢ (Tr ω → (𝑛 ∈ ω → 𝑛 ⊆ ω))
53	51, 46, 52	mpsyl 65	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω)
54	26	fdmd 5364	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:ω–onto→𝐴 → dom 𝑓 = ω)
55	54	ad2antrr 488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → dom 𝑓 = ω)
56	53, 55	sseqtrrd 3192	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → 𝑛 ⊆ dom 𝑓)
57		fores 5439	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝑓 ∧ 𝑛 ⊆ dom 𝑓) → (𝑓 ↾ 𝑛):𝑛–onto→(𝑓 “ 𝑛))
58	48, 56, 57	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → (𝑓 ↾ 𝑛):𝑛–onto→(𝑓 “ 𝑛))
59		vex 2738	. . . . . . . . . . . . . . . . 17 ⊢ 𝑓 ∈ V
60	59	resex 4941	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ↾ 𝑛) ∈ V
61		foeq1 5426	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑓 ↾ 𝑛) → (𝑔:𝑛–onto→(𝑓 “ 𝑛) ↔ (𝑓 ↾ 𝑛):𝑛–onto→(𝑓 “ 𝑛)))
62	60, 61	spcev 2830	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ↾ 𝑛):𝑛–onto→(𝑓 “ 𝑛) → ∃𝑔 𝑔:𝑛–onto→(𝑓 “ 𝑛))
63	58, 62	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → ∃𝑔 𝑔:𝑛–onto→(𝑓 “ 𝑛))
64		foeq2 5427	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑛 → (𝑔:𝑚–onto→(𝑓 “ 𝑛) ↔ 𝑔:𝑛–onto→(𝑓 “ 𝑛)))
65	64	exbidv 1823	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (∃𝑔 𝑔:𝑚–onto→(𝑓 “ 𝑛) ↔ ∃𝑔 𝑔:𝑛–onto→(𝑓 “ 𝑛)))
66	65	rspcev 2839	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ω ∧ ∃𝑔 𝑔:𝑛–onto→(𝑓 “ 𝑛)) → ∃𝑚 ∈ ω ∃𝑔 𝑔:𝑚–onto→(𝑓 “ 𝑛))
67	46, 63, 66	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → ∃𝑚 ∈ ω ∃𝑔 𝑔:𝑚–onto→(𝑓 “ 𝑛))
68	67	3adantl1 1153	. . . . . . . . . . . 12 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → ∃𝑚 ∈ ω ∃𝑔 𝑔:𝑚–onto→(𝑓 “ 𝑛))
69		fidcenum 6945	. . . . . . . . . . . 12 ⊢ ((𝑓 “ 𝑛) ∈ Fin ↔ (∀𝑎 ∈ (𝑓 “ 𝑛)∀𝑏 ∈ (𝑓 “ 𝑛)DECID 𝑎 = 𝑏 ∧ ∃𝑚 ∈ ω ∃𝑔 𝑔:𝑚–onto→(𝑓 “ 𝑛)))
70	45, 68, 69	sylanbrc 417	. . . . . . . . . . 11 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → (𝑓 “ 𝑛) ∈ Fin)
71	24, 25, 32, 70	inffinp1 12395	. . . . . . . . . 10 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → ∃𝑢 ∈ 𝐴 ¬ 𝑢 ∈ (𝑓 “ 𝑛))
72		simprl 529	. . . . . . . . . . . 12 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) ∧ (𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ∈ (𝑓 “ 𝑛))) → 𝑢 ∈ 𝐴)
73		foelrn 5744	. . . . . . . . . . . 12 ⊢ ((𝑓:ω–onto→𝐴 ∧ 𝑢 ∈ 𝐴) → ∃𝑘 ∈ ω 𝑢 = (𝑓‘𝑘))
74	33, 72, 73	syl2an2r 595	. . . . . . . . . . 11 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) ∧ (𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ∈ (𝑓 “ 𝑛))) → ∃𝑘 ∈ ω 𝑢 = (𝑓‘𝑘))
75		simpr 110	. . . . . . . . . . . . . 14 ⊢ ((((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) ∧ (𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ∈ (𝑓 “ 𝑛))) ∧ 𝑘 ∈ ω) ∧ 𝑢 = (𝑓‘𝑘)) → 𝑢 = (𝑓‘𝑘))
76		simprr 531	. . . . . . . . . . . . . . 15 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) ∧ (𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ∈ (𝑓 “ 𝑛))) → ¬ 𝑢 ∈ (𝑓 “ 𝑛))
77	76	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ ((((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) ∧ (𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ∈ (𝑓 “ 𝑛))) ∧ 𝑘 ∈ ω) ∧ 𝑢 = (𝑓‘𝑘)) → ¬ 𝑢 ∈ (𝑓 “ 𝑛))
78	75, 77	eqneltrrd 2272	. . . . . . . . . . . . 13 ⊢ ((((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) ∧ (𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ∈ (𝑓 “ 𝑛))) ∧ 𝑘 ∈ ω) ∧ 𝑢 = (𝑓‘𝑘)) → ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛))
79	78	ex 115	. . . . . . . . . . . 12 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) ∧ (𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ∈ (𝑓 “ 𝑛))) ∧ 𝑘 ∈ ω) → (𝑢 = (𝑓‘𝑘) → ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛)))
80	79	reximdva 2577	. . . . . . . . . . 11 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) ∧ (𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ∈ (𝑓 “ 𝑛))) → (∃𝑘 ∈ ω 𝑢 = (𝑓‘𝑘) → ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛)))
81	74, 80	mpd 13	. . . . . . . . . 10 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) ∧ (𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ∈ (𝑓 “ 𝑛))) → ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛))
82	71, 81	rexlimddv 2597	. . . . . . . . 9 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛))
83	82	ralrimiva 2548	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛))
84	18, 83	jca 306	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) → (𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛)))
85	84	3com23 1209	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝑓:ω–onto→𝐴) → (𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛)))
86	85	3expia 1205	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) → (𝑓:ω–onto→𝐴 → (𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛))))
87	86	eximdv 1878	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) → (∃𝑓 𝑓:ω–onto→𝐴 → ∃𝑓(𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛))))
88	16, 17, 87	sylc 62	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) → ∃𝑓(𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛)))
89	15, 88, 1	sylanbrc 417	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) → 𝐴 ≈ ℕ)
90	14, 89	impbii 126	1 ⊢ (𝐴 ≈ ℕ ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 834   ∧ w3a 978   = wceq 1353  ∃wex 1490   ∈ wcel 2146  ∀wral 2453  ∃wrex 2454   ⊆ wss 3127   class class class wbr 3998  Tr wtr 4096  Ord word 4356  ωcom 4583  dom cdm 4620  ran crn 4621   ↾ cres 4622   “ cima 4623  Fun wfun 5202  ⟶wf 5204  –onto→wfo 5206  ‘cfv 5208   ≈ cen 6728   ≼ cdom 6729  Fincfn 6730  ℕcn 8890

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-0id 7894  ax-rnegex 7895  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-ltadd 7902

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 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-1o 6407  df-er 6525  df-pm 6641  df-en 6731  df-dom 6732  df-fin 6733  df-dju 7027  df-inl 7036  df-inr 7037  df-case 7073  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-inn 8891  df-n0 9148  df-z 9225  df-uz 9500  df-fz 9978  df-seqfrec 10414

This theorem is referenced by:  qnnen  12397  unbendc  12420