Theorem ctinf 11788

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 11786	. . . 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 270	. . 3 ⊢ (𝐴 ≈ ℕ → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
3	1	simprbi 271	. . . 4 ⊢ (𝐴 ≈ ℕ → ∃𝑓(𝑓:ω–onto→𝐴 ∧ ∀𝑛 ∈ ω ∃𝑘 ∈ ω ¬ (𝑓‘𝑘) ∈ (𝑓 “ 𝑛)))
4		simpl 108	. . . . . 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 1834	. . . 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 10100	. . . . . 6 ⊢ ℕ ≈ ω
9		entr 6632	. . . . . 6 ⊢ ((𝐴 ≈ ℕ ∧ ℕ ≈ ω) → 𝐴 ≈ ω)
10	8, 9	mpan2 419	. . . . 5 ⊢ (𝐴 ≈ ℕ → 𝐴 ≈ ω)
11	10	ensymd 6631	. . . 4 ⊢ (𝐴 ≈ ℕ → ω ≈ 𝐴)
12		endom 6611	. . . 4 ⊢ (ω ≈ 𝐴 → ω ≼ 𝐴)
13	11, 12	syl 14	. . 3 ⊢ (𝐴 ≈ ℕ → ω ≼ 𝐴)
14	2, 7, 13	3jca 1144	. 2 ⊢ (𝐴 ≈ ℕ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴))
15		simp1 964	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
16		3simpb 962	. . . 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 965	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) → ∃𝑓 𝑓:ω–onto→𝐴)
18		simp2 965	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) → 𝑓:ω–onto→𝐴)
19		simpl1 967	. . . . . . . . . . . 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 1671	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑢 → (𝑥 = 𝑦 ↔ 𝑢 = 𝑦))
21	20	dcbid 806	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑢 = 𝑦))
22	21	ralbidv 2411	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → (∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ↔ ∀𝑦 ∈ 𝐴 DECID 𝑢 = 𝑦))
23	22	cbvralv 2628	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ↔ ∀𝑢 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑢 = 𝑦)
24	19, 23	sylib 121	. . . . . . . . . . 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 969	. . . . . . . . . . 11 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → ω ≼ 𝐴)
26		fof 5303	. . . . . . . . . . . . . 14 ⊢ (𝑓:ω–onto→𝐴 → 𝑓:ω⟶𝐴)
27		imassrn 4850	. . . . . . . . . . . . . . 15 ⊢ (𝑓 “ 𝑛) ⊆ ran 𝑓
28		frn 5239	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ω⟶𝐴 → ran 𝑓 ⊆ 𝐴)
29	27, 28	sstrid 3074	. . . . . . . . . . . . . 14 ⊢ (𝑓:ω⟶𝐴 → (𝑓 “ 𝑛) ⊆ 𝐴)
30	26, 29	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑓:ω–onto→𝐴 → (𝑓 “ 𝑛) ⊆ 𝐴)
31	30	ad2antrr 477	. . . . . . . . . . . 12 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → (𝑓 “ 𝑛) ⊆ 𝐴)
32	31	3adantl1 1120	. . . . . . . . . . 11 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → (𝑓 “ 𝑛) ⊆ 𝐴)
33		simpl2 968	. . . . . . . . . . . . 13 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → 𝑓:ω–onto→𝐴)
34		equequ1 1671	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → (𝑥 = 𝑦 ↔ 𝑎 = 𝑦))
35	34	dcbid 806	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑎 = 𝑦))
36		equequ2 1672	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑏 → (𝑎 = 𝑦 ↔ 𝑎 = 𝑏))
37	36	dcbid 806	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑏 → (DECID 𝑎 = 𝑦 ↔ DECID 𝑎 = 𝑏))
38	35, 37	cbvral2v 2636	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 DECID 𝑎 = 𝑏)
39		ssralv 3127	. . . . . . . . . . . . . . . . 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 2474	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ω–onto→𝐴 → (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 DECID 𝑎 = 𝑏 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ (𝑓 “ 𝑛)DECID 𝑎 = 𝑏))
42		ssralv 3127	. . . . . . . . . . . . . . 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	syl5bi 151	. . . . . . . . . . . . 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 109	. . . . . . . . . . . . . 14 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → 𝑛 ∈ ω)
47		fofun 5304	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:ω–onto→𝐴 → Fun 𝑓)
48	47	ad2antrr 477	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → Fun 𝑓)
49		ordom 4480	. . . . . . . . . . . . . . . . . . 19 ⊢ Ord ω
50		ordtr 4260	. . . . . . . . . . . . . . . . . . 19 ⊢ (Ord ω → Tr ω)
51	49, 50	ax-mp 7	. . . . . . . . . . . . . . . . . 18 ⊢ Tr ω
52		trss 3995	. . . . . . . . . . . . . . . . . 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 5237	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:ω–onto→𝐴 → dom 𝑓 = ω)
55	54	ad2antrr 477	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → dom 𝑓 = ω)
56	53, 55	sseqtr4d 3102	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → 𝑛 ⊆ dom 𝑓)
57		fores 5312	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝑓 ∧ 𝑛 ⊆ dom 𝑓) → (𝑓 ↾ 𝑛):𝑛–onto→(𝑓 “ 𝑛))
58	48, 56, 57	syl2anc 406	. . . . . . . . . . . . . . 15 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → (𝑓 ↾ 𝑛):𝑛–onto→(𝑓 “ 𝑛))
59		vex 2660	. . . . . . . . . . . . . . . . 17 ⊢ 𝑓 ∈ V
60	59	resex 4818	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ↾ 𝑛) ∈ V
61		foeq1 5299	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑓 ↾ 𝑛) → (𝑔:𝑛–onto→(𝑓 “ 𝑛) ↔ (𝑓 ↾ 𝑛):𝑛–onto→(𝑓 “ 𝑛)))
62	60, 61	spcev 2751	. . . . . . . . . . . . . . 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 5300	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑛 → (𝑔:𝑚–onto→(𝑓 “ 𝑛) ↔ 𝑔:𝑛–onto→(𝑓 “ 𝑛)))
65	64	exbidv 1779	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (∃𝑔 𝑔:𝑚–onto→(𝑓 “ 𝑛) ↔ ∃𝑔 𝑔:𝑛–onto→(𝑓 “ 𝑛)))
66	65	rspcev 2760	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ω ∧ ∃𝑔 𝑔:𝑛–onto→(𝑓 “ 𝑛)) → ∃𝑚 ∈ ω ∃𝑔 𝑔:𝑚–onto→(𝑓 “ 𝑛))
67	46, 63, 66	syl2anc 406	. . . . . . . . . . . . 13 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → ∃𝑚 ∈ ω ∃𝑔 𝑔:𝑚–onto→(𝑓 “ 𝑛))
68	67	3adantl1 1120	. . . . . . . . . . . 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 6796	. . . . . . . . . . . 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 411	. . . . . . . . . . 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 11787	. . . . . . . . . 10 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → ∃𝑢 ∈ 𝐴 ¬ 𝑢 ∈ (𝑓 “ 𝑛))
72		simprl 503	. . . . . . . . . . . 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 5608	. . . . . . . . . . . 12 ⊢ ((𝑓:ω–onto→𝐴 ∧ 𝑢 ∈ 𝐴) → ∃𝑘 ∈ ω 𝑢 = (𝑓‘𝑘))
74	33, 72, 73	syl2an2r 567	. . . . . . . . . . 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 109	. . . . . . . . . . . . . 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 504	. . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . 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 2211	. . . . . . . . . . . . 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 114	. . . . . . . . . . . 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 2508	. . . . . . . . . . 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 2528	. . . . . . . . 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 2479	. . . . . . . 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 302	. . . . . . 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 1170	. . . . . 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 1166	. . . . 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 1834	. . . 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 411	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) → 𝐴 ≈ ℕ)
90	14, 89	impbii 125	1 ⊢ (𝐴 ≈ ℕ ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 802   ∧ w3a 945   = wceq 1314  ∃wex 1451   ∈ wcel 1463  ∀wral 2390  ∃wrex 2391   ⊆ wss 3037   class class class wbr 3895  Tr wtr 3986  Ord word 4244  ωcom 4464  dom cdm 4499  ran crn 4500   ↾ cres 4501   “ cima 4502  Fun wfun 5075  ⟶wf 5077  –onto→wfo 5079  ‘cfv 5081   ≈ cen 6586   ≼ cdom 6587  Fincfn 6588  ℕcn 8630

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-addcom 7645  ax-addass 7647  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-0id 7653  ax-rnegex 7654  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-ltadd 7661

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-iord 4248  df-on 4250  df-ilim 4251  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-frec 6242  df-1o 6267  df-er 6383  df-pm 6499  df-en 6589  df-dom 6590  df-fin 6591  df-dju 6875  df-inl 6884  df-inr 6885  df-case 6921  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-inn 8631  df-n0 8882  df-z 8959  df-uz 9229  df-fz 9684  df-seqfrec 10112

This theorem is referenced by:  qnnen  11789