Theorem ctinf 13373

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 13371	. . . 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 1933	. . . 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 10886	. . . . . 6 ⊢ ℕ ≈ ω
9		entr 7071	. . . . . 6 ⊢ ((𝐴 ≈ ℕ ∧ ℕ ≈ ω) → 𝐴 ≈ ω)
10	8, 9	mpan2 429	. . . . 5 ⊢ (𝐴 ≈ ℕ → 𝐴 ≈ ω)
11	10	ensymd 7070	. . . 4 ⊢ (𝐴 ≈ ℕ → ω ≈ 𝐴)
12		endom 7049	. . . 4 ⊢ (ω ≈ 𝐴 → ω ≼ 𝐴)
13	11, 12	syl 14	. . 3 ⊢ (𝐴 ≈ ℕ → ω ≼ 𝐴)
14	2, 7, 13	3jca 1208	. 2 ⊢ (𝐴 ≈ ℕ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴))
15		simp1 1028	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
16		3simpb 1026	. . . 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 1029	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) → ∃𝑓 𝑓:ω–onto→𝐴)
18		simp2 1029	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) → 𝑓:ω–onto→𝐴)
19		simpl1 1031	. . . . . . . . . . . 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 1764	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑢 → (𝑥 = 𝑦 ↔ 𝑢 = 𝑦))
21	20	dcbid 850	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑢 = 𝑦))
22	21	ralbidv 2550	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → (∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ↔ ∀𝑦 ∈ 𝐴 DECID 𝑢 = 𝑦))
23	22	cbvralv 2786	. . . . . . . . . . . 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 1033	. . . . . . . . . . 11 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → ω ≼ 𝐴)
26		fof 5615	. . . . . . . . . . . . . 14 ⊢ (𝑓:ω–onto→𝐴 → 𝑓:ω⟶𝐴)
27		imassrn 5137	. . . . . . . . . . . . . . 15 ⊢ (𝑓 “ 𝑛) ⊆ ran 𝑓
28		frn 5542	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ω⟶𝐴 → ran 𝑓 ⊆ 𝐴)
29	27, 28	sstrid 3259	. . . . . . . . . . . . . 14 ⊢ (𝑓:ω⟶𝐴 → (𝑓 “ 𝑛) ⊆ 𝐴)
30	26, 29	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑓:ω–onto→𝐴 → (𝑓 “ 𝑛) ⊆ 𝐴)
31	30	ad2antrr 492	. . . . . . . . . . . 12 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → (𝑓 “ 𝑛) ⊆ 𝐴)
32	31	3adantl1 1184	. . . . . . . . . . 11 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → (𝑓 “ 𝑛) ⊆ 𝐴)
33		simpl2 1032	. . . . . . . . . . . . 13 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → 𝑓:ω–onto→𝐴)
34		equequ1 1764	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → (𝑥 = 𝑦 ↔ 𝑎 = 𝑦))
35	34	dcbid 850	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑎 = 𝑦))
36		equequ2 1765	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑏 → (𝑎 = 𝑦 ↔ 𝑎 = 𝑏))
37	36	dcbid 850	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑏 → (DECID 𝑎 = 𝑦 ↔ DECID 𝑎 = 𝑏))
38	35, 37	cbvral2v 2799	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 DECID 𝑎 = 𝑏)
39		ssralv 3312	. . . . . . . . . . . . . . . . 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 2618	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ω–onto→𝐴 → (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 DECID 𝑎 = 𝑏 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ (𝑓 “ 𝑛)DECID 𝑎 = 𝑏))
42		ssralv 3312	. . . . . . . . . . . . . . 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 5616	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:ω–onto→𝐴 → Fun 𝑓)
48	47	ad2antrr 492	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → Fun 𝑓)
49		ordom 4754	. . . . . . . . . . . . . . . . . . 19 ⊢ Ord ω
50		ordtr 4523	. . . . . . . . . . . . . . . . . . 19 ⊢ (Ord ω → Tr ω)
51	49, 50	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ Tr ω
52		trss 4238	. . . . . . . . . . . . . . . . . 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 5540	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:ω–onto→𝐴 → dom 𝑓 = ω)
55	54	ad2antrr 492	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → dom 𝑓 = ω)
56	53, 55	sseqtrrd 3287	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → 𝑛 ⊆ dom 𝑓)
57		fores 5625	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝑓 ∧ 𝑛 ⊆ dom 𝑓) → (𝑓 ↾ 𝑛):𝑛–onto→(𝑓 “ 𝑛))
58	48, 56, 57	syl2anc 415	. . . . . . . . . . . . . . 15 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → (𝑓 ↾ 𝑛):𝑛–onto→(𝑓 “ 𝑛))
59		vex 2824	. . . . . . . . . . . . . . . . 17 ⊢ 𝑓 ∈ V
60	59	resex 5104	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ↾ 𝑛) ∈ V
61		foeq1 5611	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑓 ↾ 𝑛) → (𝑔:𝑛–onto→(𝑓 “ 𝑛) ↔ (𝑓 ↾ 𝑛):𝑛–onto→(𝑓 “ 𝑛)))
62	60, 61	spcev 2920	. . . . . . . . . . . . . . 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 5612	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑛 → (𝑔:𝑚–onto→(𝑓 “ 𝑛) ↔ 𝑔:𝑛–onto→(𝑓 “ 𝑛)))
65	64	exbidv 1878	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (∃𝑔 𝑔:𝑚–onto→(𝑓 “ 𝑛) ↔ ∃𝑔 𝑔:𝑛–onto→(𝑓 “ 𝑛)))
66	65	rspcev 2929	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ω ∧ ∃𝑔 𝑔:𝑛–onto→(𝑓 “ 𝑛)) → ∃𝑚 ∈ ω ∃𝑔 𝑔:𝑚–onto→(𝑓 “ 𝑛))
67	46, 63, 66	syl2anc 415	. . . . . . . . . . . . 13 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → ∃𝑚 ∈ ω ∃𝑔 𝑔:𝑚–onto→(𝑓 “ 𝑛))
68	67	3adantl1 1184	. . . . . . . . . . . 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 7273	. . . . . . . . . . . 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 421	. . . . . . . . . . 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 13372	. . . . . . . . . 10 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → ∃𝑢 ∈ 𝐴 ¬ 𝑢 ∈ (𝑓 “ 𝑛))
72		simprl 535	. . . . . . . . . . . 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 5958	. . . . . . . . . . . 12 ⊢ ((𝑓:ω–onto→𝐴 ∧ 𝑢 ∈ 𝐴) → ∃𝑘 ∈ ω 𝑢 = (𝑓‘𝑘))
74	33, 72, 73	syl2an2r 603	. . . . . . . . . . 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 537	. . . . . . . . . . . . . . 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 492	. . . . . . . . . . . . . 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 2335	. . . . . . . . . . . . 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 2652	. . . . . . . . . . 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 2673	. . . . . . . . 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 2623	. . . . . . . 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 1240	. . . . . 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 1236	. . . . 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 1933	. . . 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 421	. 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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 846   ∧ w3a 1009   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529   ⊆ wss 3220   class class class wbr 4130  Tr wtr 4229  Ord word 4507  ωcom 4737  dom cdm 4774  ran crn 4775   ↾ cres 4776   “ cima 4777  Fun wfun 5371  ⟶wf 5373  –onto→wfo 5375  ‘cfv 5377   ≈ cen 7020   ≼ cdom 7021  Fincfn 7022  ℕcn 9307

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-addcom 8280  ax-addass 8282  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-0id 8288  ax-rnegex 8289  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-ltadd 8296

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-pm 6925  df-en 7023  df-dom 7024  df-fin 7025  df-dju 7379  df-inl 7388  df-inr 7389  df-case 7425  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-inn 9308  df-n0 9569  df-z 9650  df-uz 9932  df-fz 10423  df-seqfrec 10900

This theorem is used by:  qnnen  13374  unbendc  13397  nnnninfen  17235