Theorem ctinf 13044

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 13042	. . . 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 1926	. . . 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 10689	. . . . . 6 ⊢ ℕ ≈ ω
9		entr 6953	. . . . . 6 ⊢ ((𝐴 ≈ ℕ ∧ ℕ ≈ ω) → 𝐴 ≈ ω)
10	8, 9	mpan2 425	. . . . 5 ⊢ (𝐴 ≈ ℕ → 𝐴 ≈ ω)
11	10	ensymd 6952	. . . 4 ⊢ (𝐴 ≈ ℕ → ω ≈ 𝐴)
12		endom 6931	. . . 4 ⊢ (ω ≈ 𝐴 → ω ≼ 𝐴)
13	11, 12	syl 14	. . 3 ⊢ (𝐴 ≈ ℕ → ω ≼ 𝐴)
14	2, 7, 13	3jca 1201	. 2 ⊢ (𝐴 ≈ ℕ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴))
15		simp1 1021	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
16		3simpb 1019	. . . 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 1022	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) → ∃𝑓 𝑓:ω–onto→𝐴)
18		simp2 1022	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) → 𝑓:ω–onto→𝐴)
19		simpl1 1024	. . . . . . . . . . . 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 1758	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑢 → (𝑥 = 𝑦 ↔ 𝑢 = 𝑦))
21	20	dcbid 843	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑢 = 𝑦))
22	21	ralbidv 2530	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → (∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ↔ ∀𝑦 ∈ 𝐴 DECID 𝑢 = 𝑦))
23	22	cbvralv 2765	. . . . . . . . . . . 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 1026	. . . . . . . . . . 11 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → ω ≼ 𝐴)
26		fof 5556	. . . . . . . . . . . . . 14 ⊢ (𝑓:ω–onto→𝐴 → 𝑓:ω⟶𝐴)
27		imassrn 5085	. . . . . . . . . . . . . . 15 ⊢ (𝑓 “ 𝑛) ⊆ ran 𝑓
28		frn 5488	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ω⟶𝐴 → ran 𝑓 ⊆ 𝐴)
29	27, 28	sstrid 3236	. . . . . . . . . . . . . 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 1177	. . . . . . . . . . 11 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → (𝑓 “ 𝑛) ⊆ 𝐴)
33		simpl2 1025	. . . . . . . . . . . . 13 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → 𝑓:ω–onto→𝐴)
34		equequ1 1758	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → (𝑥 = 𝑦 ↔ 𝑎 = 𝑦))
35	34	dcbid 843	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑎 = 𝑦))
36		equequ2 1759	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑏 → (𝑎 = 𝑦 ↔ 𝑎 = 𝑏))
37	36	dcbid 843	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑏 → (DECID 𝑎 = 𝑦 ↔ DECID 𝑎 = 𝑏))
38	35, 37	cbvral2v 2778	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 DECID 𝑎 = 𝑏)
39		ssralv 3289	. . . . . . . . . . . . . . . . 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 2598	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ω–onto→𝐴 → (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 DECID 𝑎 = 𝑏 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ (𝑓 “ 𝑛)DECID 𝑎 = 𝑏))
42		ssralv 3289	. . . . . . . . . . . . . . 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 5557	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:ω–onto→𝐴 → Fun 𝑓)
48	47	ad2antrr 488	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → Fun 𝑓)
49		ordom 4703	. . . . . . . . . . . . . . . . . . 19 ⊢ Ord ω
50		ordtr 4473	. . . . . . . . . . . . . . . . . . 19 ⊢ (Ord ω → Tr ω)
51	49, 50	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ Tr ω
52		trss 4194	. . . . . . . . . . . . . . . . . 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 5486	. . . . . . . . . . . . . . . . . 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 3264	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓:ω–onto→𝐴 ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → 𝑛 ⊆ dom 𝑓)
57		fores 5566	. . . . . . . . . . . . . . . 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 2803	. . . . . . . . . . . . . . . . 17 ⊢ 𝑓 ∈ V
60	59	resex 5052	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ↾ 𝑛) ∈ V
61		foeq1 5552	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑓 ↾ 𝑛) → (𝑔:𝑛–onto→(𝑓 “ 𝑛) ↔ (𝑓 ↾ 𝑛):𝑛–onto→(𝑓 “ 𝑛)))
62	60, 61	spcev 2899	. . . . . . . . . . . . . . 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 5553	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑛 → (𝑔:𝑚–onto→(𝑓 “ 𝑛) ↔ 𝑔:𝑛–onto→(𝑓 “ 𝑛)))
65	64	exbidv 1871	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (∃𝑔 𝑔:𝑚–onto→(𝑓 “ 𝑛) ↔ ∃𝑔 𝑔:𝑛–onto→(𝑓 “ 𝑛)))
66	65	rspcev 2908	. . . . . . . . . . . . . 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 1177	. . . . . . . . . . . 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 7149	. . . . . . . . . . . 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 13043	. . . . . . . . . 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 5888	. . . . . . . . . . . 12 ⊢ ((𝑓:ω–onto→𝐴 ∧ 𝑢 ∈ 𝐴) → ∃𝑘 ∈ ω 𝑢 = (𝑓‘𝑘))
74	33, 72, 73	syl2an2r 597	. . . . . . . . . . 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 2326	. . . . . . . . . . . . 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 2632	. . . . . . . . . . 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 2653	. . . . . . . . 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 2603	. . . . . . . 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 1233	. . . . . 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 1229	. . . . 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 1926	. . . 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 839   ∧ w3a 1002   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509   ⊆ wss 3198   class class class wbr 4086  Tr wtr 4185  Ord word 4457  ωcom 4686  dom cdm 4723  ran crn 4724   ↾ cres 4725   “ cima 4726  Fun wfun 5318  ⟶wf 5320  –onto→wfo 5322  ‘cfv 5324   ≈ cen 6902   ≼ cdom 6903  Fincfn 6904  ℕcn 9136

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-addass 8127  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-0id 8133  ax-rnegex 8134  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-ltadd 8141

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-1o 6577  df-er 6697  df-pm 6815  df-en 6905  df-dom 6906  df-fin 6907  df-dju 7231  df-inl 7240  df-inr 7241  df-case 7277  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-inn 9137  df-n0 9396  df-z 9473  df-uz 9749  df-fz 10237  df-seqfrec 10703

This theorem is referenced by:  qnnen  13045  unbendc  13068  nnnninfen  16573