Theorem ennnfonelemim 12379

Description: Lemma for ennnfone 12380. The trivial direction. (Contributed by Jim Kingdon, 27-Oct-2022.)

Assertion

Ref	Expression
ennnfonelemim	⊢ (𝐴 ≈ ℕ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ℕ0–onto→𝐴 ∧ ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝑓‘𝑘) ≠ (𝑓‘𝑗))))

Distinct variable groups:   𝐴,𝑓,𝑗,𝑛   𝑥,𝐴,𝑦,𝑛   𝑓,𝑘,𝑗,𝑛   𝑦,𝑗

Allowed substitution hint:   𝐴(𝑘)

Proof of Theorem ennnfonelemim

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0ennn 10389	. . . 4 ⊢ ℕ0 ≈ ℕ
2	1	ensymi 6760	. . 3 ⊢ ℕ ≈ ℕ0
3		entr 6762	. . 3 ⊢ ((𝐴 ≈ ℕ ∧ ℕ ≈ ℕ0) → 𝐴 ≈ ℕ0)
4	2, 3	mpan2 423	. 2 ⊢ (𝐴 ≈ ℕ → 𝐴 ≈ ℕ0)
5		bren 6725	. . . 4 ⊢ (𝐴 ≈ ℕ0 ↔ ∃𝑔 𝑔:𝐴–1-1-onto→ℕ0)
6	5	biimpi 119	. . 3 ⊢ (𝐴 ≈ ℕ0 → ∃𝑔 𝑔:𝐴–1-1-onto→ℕ0)
7		f1of 5442	. . . . . . . . . . 11 ⊢ (𝑔:𝐴–1-1-onto→ℕ0 → 𝑔:𝐴⟶ℕ0)
8	7	adantr 274	. . . . . . . . . 10 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑔:𝐴⟶ℕ0)
9		simprl 526	. . . . . . . . . 10 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
10	8, 9	ffvelrnd 5632	. . . . . . . . 9 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑔‘𝑥) ∈ ℕ0)
11	10	nn0zd 9332	. . . . . . . 8 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑔‘𝑥) ∈ ℤ)
12		simprr 527	. . . . . . . . . 10 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
13	8, 12	ffvelrnd 5632	. . . . . . . . 9 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑔‘𝑦) ∈ ℕ0)
14	13	nn0zd 9332	. . . . . . . 8 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑔‘𝑦) ∈ ℤ)
15		zdceq 9287	. . . . . . . 8 ⊢ (((𝑔‘𝑥) ∈ ℤ ∧ (𝑔‘𝑦) ∈ ℤ) → DECID (𝑔‘𝑥) = (𝑔‘𝑦))
16	11, 14, 15	syl2anc 409	. . . . . . 7 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → DECID (𝑔‘𝑥) = (𝑔‘𝑦))
17		dff1o6 5755	. . . . . . . . . . . . 13 ⊢ (𝑔:𝐴–1-1-onto→ℕ0 ↔ (𝑔 Fn 𝐴 ∧ ran 𝑔 = ℕ0 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑔‘𝑥) = (𝑔‘𝑦) → 𝑥 = 𝑦)))
18	17	simp3bi 1009	. . . . . . . . . . . 12 ⊢ (𝑔:𝐴–1-1-onto→ℕ0 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑔‘𝑥) = (𝑔‘𝑦) → 𝑥 = 𝑦))
19	18	r19.21bi 2558	. . . . . . . . . . 11 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 ((𝑔‘𝑥) = (𝑔‘𝑦) → 𝑥 = 𝑦))
20	19	r19.21bi 2558	. . . . . . . . . 10 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝑔‘𝑥) = (𝑔‘𝑦) → 𝑥 = 𝑦))
21	20	anasss 397	. . . . . . . . 9 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑔‘𝑥) = (𝑔‘𝑦) → 𝑥 = 𝑦))
22		fveq2 5496	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑔‘𝑥) = (𝑔‘𝑦))
23	21, 22	impbid1 141	. . . . . . . 8 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑔‘𝑥) = (𝑔‘𝑦) ↔ 𝑥 = 𝑦))
24	23	dcbid 833	. . . . . . 7 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (DECID (𝑔‘𝑥) = (𝑔‘𝑦) ↔ DECID 𝑥 = 𝑦))
25	16, 24	mpbid 146	. . . . . 6 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → DECID 𝑥 = 𝑦)
26	25	ralrimivva 2552	. . . . 5 ⊢ (𝑔:𝐴–1-1-onto→ℕ0 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
27		f1ocnv 5455	. . . . . . 7 ⊢ (𝑔:𝐴–1-1-onto→ℕ0 → ◡𝑔:ℕ0–1-1-onto→𝐴)
28		f1ofo 5449	. . . . . . 7 ⊢ (◡𝑔:ℕ0–1-1-onto→𝐴 → ◡𝑔:ℕ0–onto→𝐴)
29	27, 28	syl 14	. . . . . 6 ⊢ (𝑔:𝐴–1-1-onto→ℕ0 → ◡𝑔:ℕ0–onto→𝐴)
30		peano2nn0 9175	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
31	30	adantl 275	. . . . . . . 8 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) → (𝑛 + 1) ∈ ℕ0)
32		elfznn0 10070	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...𝑛) → 𝑗 ∈ ℕ0)
33	32	adantl 275	. . . . . . . . . . . . . 14 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → 𝑗 ∈ ℕ0)
34	33	nn0red 9189	. . . . . . . . . . . . 13 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → 𝑗 ∈ ℝ)
35		elfzle2 9984	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...𝑛) → 𝑗 ≤ 𝑛)
36	35	adantl 275	. . . . . . . . . . . . . 14 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → 𝑗 ≤ 𝑛)
37		simplr 525	. . . . . . . . . . . . . . 15 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → 𝑛 ∈ ℕ0)
38		nn0leltp1 9275	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → (𝑗 ≤ 𝑛 ↔ 𝑗 < (𝑛 + 1)))
39	33, 37, 38	syl2anc 409	. . . . . . . . . . . . . 14 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → (𝑗 ≤ 𝑛 ↔ 𝑗 < (𝑛 + 1)))
40	36, 39	mpbid 146	. . . . . . . . . . . . 13 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → 𝑗 < (𝑛 + 1))
41	34, 40	gtned 8032	. . . . . . . . . . . 12 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → (𝑛 + 1) ≠ 𝑗)
42	41	neneqd 2361	. . . . . . . . . . 11 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → ¬ (𝑛 + 1) = 𝑗)
43		dff1o6 5755	. . . . . . . . . . . . . . 15 ⊢ (◡𝑔:ℕ0–1-1-onto→𝐴 ↔ (◡𝑔 Fn ℕ0 ∧ ran ◡𝑔 = 𝐴 ∧ ∀𝑥 ∈ ℕ0 ∀𝑦 ∈ ℕ0 ((◡𝑔‘𝑥) = (◡𝑔‘𝑦) → 𝑥 = 𝑦)))
44	27, 43	sylib 121	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝐴–1-1-onto→ℕ0 → (◡𝑔 Fn ℕ0 ∧ ran ◡𝑔 = 𝐴 ∧ ∀𝑥 ∈ ℕ0 ∀𝑦 ∈ ℕ0 ((◡𝑔‘𝑥) = (◡𝑔‘𝑦) → 𝑥 = 𝑦)))
45	44	simp3d 1006	. . . . . . . . . . . . 13 ⊢ (𝑔:𝐴–1-1-onto→ℕ0 → ∀𝑥 ∈ ℕ0 ∀𝑦 ∈ ℕ0 ((◡𝑔‘𝑥) = (◡𝑔‘𝑦) → 𝑥 = 𝑦))
46	45	ad2antrr 485	. . . . . . . . . . . 12 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑥 ∈ ℕ0 ∀𝑦 ∈ ℕ0 ((◡𝑔‘𝑥) = (◡𝑔‘𝑦) → 𝑥 = 𝑦))
47	31	adantr 274	. . . . . . . . . . . . 13 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → (𝑛 + 1) ∈ ℕ0)
48		fveqeq2 5505	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑛 + 1) → ((◡𝑔‘𝑥) = (◡𝑔‘𝑦) ↔ (◡𝑔‘(𝑛 + 1)) = (◡𝑔‘𝑦)))
49		eqeq1 2177	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 = 𝑦 ↔ (𝑛 + 1) = 𝑦))
50	48, 49	imbi12d 233	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑛 + 1) → (((◡𝑔‘𝑥) = (◡𝑔‘𝑦) → 𝑥 = 𝑦) ↔ ((◡𝑔‘(𝑛 + 1)) = (◡𝑔‘𝑦) → (𝑛 + 1) = 𝑦)))
51		fveq2 5496	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑗 → (◡𝑔‘𝑦) = (◡𝑔‘𝑗))
52	51	eqeq2d 2182	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑗 → ((◡𝑔‘(𝑛 + 1)) = (◡𝑔‘𝑦) ↔ (◡𝑔‘(𝑛 + 1)) = (◡𝑔‘𝑗)))
53		eqeq2 2180	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑗 → ((𝑛 + 1) = 𝑦 ↔ (𝑛 + 1) = 𝑗))
54	52, 53	imbi12d 233	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑗 → (((◡𝑔‘(𝑛 + 1)) = (◡𝑔‘𝑦) → (𝑛 + 1) = 𝑦) ↔ ((◡𝑔‘(𝑛 + 1)) = (◡𝑔‘𝑗) → (𝑛 + 1) = 𝑗)))
55	50, 54	rspc2v 2847	. . . . . . . . . . . . 13 ⊢ (((𝑛 + 1) ∈ ℕ0 ∧ 𝑗 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 ∀𝑦 ∈ ℕ0 ((◡𝑔‘𝑥) = (◡𝑔‘𝑦) → 𝑥 = 𝑦) → ((◡𝑔‘(𝑛 + 1)) = (◡𝑔‘𝑗) → (𝑛 + 1) = 𝑗)))
56	47, 33, 55	syl2anc 409	. . . . . . . . . . . 12 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → (∀𝑥 ∈ ℕ0 ∀𝑦 ∈ ℕ0 ((◡𝑔‘𝑥) = (◡𝑔‘𝑦) → 𝑥 = 𝑦) → ((◡𝑔‘(𝑛 + 1)) = (◡𝑔‘𝑗) → (𝑛 + 1) = 𝑗)))
57	46, 56	mpd 13	. . . . . . . . . . 11 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → ((◡𝑔‘(𝑛 + 1)) = (◡𝑔‘𝑗) → (𝑛 + 1) = 𝑗))
58	42, 57	mtod 658	. . . . . . . . . 10 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → ¬ (◡𝑔‘(𝑛 + 1)) = (◡𝑔‘𝑗))
59	58	neqned 2347	. . . . . . . . 9 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → (◡𝑔‘(𝑛 + 1)) ≠ (◡𝑔‘𝑗))
60	59	ralrimiva 2543	. . . . . . . 8 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) → ∀𝑗 ∈ (0...𝑛)(◡𝑔‘(𝑛 + 1)) ≠ (◡𝑔‘𝑗))
61		fveq2 5496	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑛 + 1) → (◡𝑔‘𝑘) = (◡𝑔‘(𝑛 + 1)))
62	61	neeq1d 2358	. . . . . . . . . 10 ⊢ (𝑘 = (𝑛 + 1) → ((◡𝑔‘𝑘) ≠ (◡𝑔‘𝑗) ↔ (◡𝑔‘(𝑛 + 1)) ≠ (◡𝑔‘𝑗)))
63	62	ralbidv 2470	. . . . . . . . 9 ⊢ (𝑘 = (𝑛 + 1) → (∀𝑗 ∈ (0...𝑛)(◡𝑔‘𝑘) ≠ (◡𝑔‘𝑗) ↔ ∀𝑗 ∈ (0...𝑛)(◡𝑔‘(𝑛 + 1)) ≠ (◡𝑔‘𝑗)))
64	63	rspcev 2834	. . . . . . . 8 ⊢ (((𝑛 + 1) ∈ ℕ0 ∧ ∀𝑗 ∈ (0...𝑛)(◡𝑔‘(𝑛 + 1)) ≠ (◡𝑔‘𝑗)) → ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(◡𝑔‘𝑘) ≠ (◡𝑔‘𝑗))
65	31, 60, 64	syl2anc 409	. . . . . . 7 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) → ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(◡𝑔‘𝑘) ≠ (◡𝑔‘𝑗))
66	65	ralrimiva 2543	. . . . . 6 ⊢ (𝑔:𝐴–1-1-onto→ℕ0 → ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(◡𝑔‘𝑘) ≠ (◡𝑔‘𝑗))
67		cnvexg 5148	. . . . . . . 8 ⊢ (𝑔 ∈ V → ◡𝑔 ∈ V)
68	67	elv 2734	. . . . . . 7 ⊢ ◡𝑔 ∈ V
69		foeq1 5416	. . . . . . . 8 ⊢ (𝑓 = ◡𝑔 → (𝑓:ℕ0–onto→𝐴 ↔ ◡𝑔:ℕ0–onto→𝐴))
70		fveq1 5495	. . . . . . . . . . 11 ⊢ (𝑓 = ◡𝑔 → (𝑓‘𝑘) = (◡𝑔‘𝑘))
71		fveq1 5495	. . . . . . . . . . 11 ⊢ (𝑓 = ◡𝑔 → (𝑓‘𝑗) = (◡𝑔‘𝑗))
72	70, 71	neeq12d 2360	. . . . . . . . . 10 ⊢ (𝑓 = ◡𝑔 → ((𝑓‘𝑘) ≠ (𝑓‘𝑗) ↔ (◡𝑔‘𝑘) ≠ (◡𝑔‘𝑗)))
73	72	rexralbidv 2496	. . . . . . . . 9 ⊢ (𝑓 = ◡𝑔 → (∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝑓‘𝑘) ≠ (𝑓‘𝑗) ↔ ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(◡𝑔‘𝑘) ≠ (◡𝑔‘𝑗)))
74	73	ralbidv 2470	. . . . . . . 8 ⊢ (𝑓 = ◡𝑔 → (∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝑓‘𝑘) ≠ (𝑓‘𝑗) ↔ ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(◡𝑔‘𝑘) ≠ (◡𝑔‘𝑗)))
75	69, 74	anbi12d 470	. . . . . . 7 ⊢ (𝑓 = ◡𝑔 → ((𝑓:ℕ0–onto→𝐴 ∧ ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝑓‘𝑘) ≠ (𝑓‘𝑗)) ↔ (◡𝑔:ℕ0–onto→𝐴 ∧ ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(◡𝑔‘𝑘) ≠ (◡𝑔‘𝑗))))
76	68, 75	spcev 2825	. . . . . 6 ⊢ ((◡𝑔:ℕ0–onto→𝐴 ∧ ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(◡𝑔‘𝑘) ≠ (◡𝑔‘𝑗)) → ∃𝑓(𝑓:ℕ0–onto→𝐴 ∧ ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝑓‘𝑘) ≠ (𝑓‘𝑗)))
77	29, 66, 76	syl2anc 409	. . . . 5 ⊢ (𝑔:𝐴–1-1-onto→ℕ0 → ∃𝑓(𝑓:ℕ0–onto→𝐴 ∧ ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝑓‘𝑘) ≠ (𝑓‘𝑗)))
78	26, 77	jca 304	. . . 4 ⊢ (𝑔:𝐴–1-1-onto→ℕ0 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ℕ0–onto→𝐴 ∧ ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝑓‘𝑘) ≠ (𝑓‘𝑗))))
79	78	adantl 275	. . 3 ⊢ ((𝐴 ≈ ℕ0 ∧ 𝑔:𝐴–1-1-onto→ℕ0) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ℕ0–onto→𝐴 ∧ ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝑓‘𝑘) ≠ (𝑓‘𝑗))))
80	6, 79	exlimddv 1891	. 2 ⊢ (𝐴 ≈ ℕ0 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ℕ0–onto→𝐴 ∧ ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝑓‘𝑘) ≠ (𝑓‘𝑗))))
81	4, 80	syl 14	1 ⊢ (𝐴 ≈ ℕ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ℕ0–onto→𝐴 ∧ ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝑓‘𝑘) ≠ (𝑓‘𝑗))))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 829   ∧ w3a 973   = wceq 1348  ∃wex 1485   ∈ wcel 2141   ≠ wne 2340  ∀wral 2448  ∃wrex 2449  Vcvv 2730   class class class wbr 3989  ^◡ccnv 4610  ran crn 4612   Fn wfn 5193  ⟶wf 5194  –onto→wfo 5196  –1-1-onto→wf1o 5197  ‘cfv 5198  (class class class)co 5853   ≈ cen 6716  0cc0 7774  1c1 7775   + caddc 7777   < clt 7954   ≤ cle 7955  ℕcn 8878  ℕ₀cn0 9135  ℤcz 9212  ...cfz 9965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-er 6513  df-en 6719  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966

This theorem is referenced by:  ennnfone  12380