Theorem ennnfonelemim 12910

Description: Lemma for ennnfone 12911. 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 10615	. . . 4 ⊢ ℕ0 ≈ ℕ
2	1	ensymi 6897	. . 3 ⊢ ℕ ≈ ℕ0
3		entr 6899	. . 3 ⊢ ((𝐴 ≈ ℕ ∧ ℕ ≈ ℕ0) → 𝐴 ≈ ℕ0)
4	2, 3	mpan2 425	. 2 ⊢ (𝐴 ≈ ℕ → 𝐴 ≈ ℕ0)
5		bren 6858	. . . 4 ⊢ (𝐴 ≈ ℕ0 ↔ ∃𝑔 𝑔:𝐴–1-1-onto→ℕ0)
6	5	biimpi 120	. . 3 ⊢ (𝐴 ≈ ℕ0 → ∃𝑔 𝑔:𝐴–1-1-onto→ℕ0)
7		f1of 5544	. . . . . . . . . . 11 ⊢ (𝑔:𝐴–1-1-onto→ℕ0 → 𝑔:𝐴⟶ℕ0)
8	7	adantr 276	. . . . . . . . . 10 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑔:𝐴⟶ℕ0)
9		simprl 529	. . . . . . . . . 10 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
10	8, 9	ffvelcdmd 5739	. . . . . . . . 9 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑔‘𝑥) ∈ ℕ0)
11	10	nn0zd 9528	. . . . . . . 8 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑔‘𝑥) ∈ ℤ)
12		simprr 531	. . . . . . . . . 10 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
13	8, 12	ffvelcdmd 5739	. . . . . . . . 9 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑔‘𝑦) ∈ ℕ0)
14	13	nn0zd 9528	. . . . . . . 8 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑔‘𝑦) ∈ ℤ)
15		zdceq 9483	. . . . . . . 8 ⊢ (((𝑔‘𝑥) ∈ ℤ ∧ (𝑔‘𝑦) ∈ ℤ) → DECID (𝑔‘𝑥) = (𝑔‘𝑦))
16	11, 14, 15	syl2anc 411	. . . . . . 7 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → DECID (𝑔‘𝑥) = (𝑔‘𝑦))
17		dff1o6 5868	. . . . . . . . . . . . 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 1017	. . . . . . . . . . . 12 ⊢ (𝑔:𝐴–1-1-onto→ℕ0 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑔‘𝑥) = (𝑔‘𝑦) → 𝑥 = 𝑦))
19	18	r19.21bi 2596	. . . . . . . . . . 11 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 ((𝑔‘𝑥) = (𝑔‘𝑦) → 𝑥 = 𝑦))
20	19	r19.21bi 2596	. . . . . . . . . 10 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝑔‘𝑥) = (𝑔‘𝑦) → 𝑥 = 𝑦))
21	20	anasss 399	. . . . . . . . 9 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑔‘𝑥) = (𝑔‘𝑦) → 𝑥 = 𝑦))
22		fveq2 5599	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑔‘𝑥) = (𝑔‘𝑦))
23	21, 22	impbid1 142	. . . . . . . 8 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑔‘𝑥) = (𝑔‘𝑦) ↔ 𝑥 = 𝑦))
24	23	dcbid 840	. . . . . . 7 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (DECID (𝑔‘𝑥) = (𝑔‘𝑦) ↔ DECID 𝑥 = 𝑦))
25	16, 24	mpbid 147	. . . . . 6 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → DECID 𝑥 = 𝑦)
26	25	ralrimivva 2590	. . . . 5 ⊢ (𝑔:𝐴–1-1-onto→ℕ0 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
27		f1ocnv 5557	. . . . . . 7 ⊢ (𝑔:𝐴–1-1-onto→ℕ0 → ◡𝑔:ℕ0–1-1-onto→𝐴)
28		f1ofo 5551	. . . . . . 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 9370	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
31	30	adantl 277	. . . . . . . 8 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) → (𝑛 + 1) ∈ ℕ0)
32		elfznn0 10271	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...𝑛) → 𝑗 ∈ ℕ0)
33	32	adantl 277	. . . . . . . . . . . . . 14 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → 𝑗 ∈ ℕ0)
34	33	nn0red 9384	. . . . . . . . . . . . 13 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → 𝑗 ∈ ℝ)
35		elfzle2 10185	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...𝑛) → 𝑗 ≤ 𝑛)
36	35	adantl 277	. . . . . . . . . . . . . 14 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → 𝑗 ≤ 𝑛)
37		simplr 528	. . . . . . . . . . . . . . 15 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → 𝑛 ∈ ℕ0)
38		nn0leltp1 9471	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → (𝑗 ≤ 𝑛 ↔ 𝑗 < (𝑛 + 1)))
39	33, 37, 38	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → (𝑗 ≤ 𝑛 ↔ 𝑗 < (𝑛 + 1)))
40	36, 39	mpbid 147	. . . . . . . . . . . . 13 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → 𝑗 < (𝑛 + 1))
41	34, 40	gtned 8220	. . . . . . . . . . . 12 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → (𝑛 + 1) ≠ 𝑗)
42	41	neneqd 2399	. . . . . . . . . . 11 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → ¬ (𝑛 + 1) = 𝑗)
43		dff1o6 5868	. . . . . . . . . . . . . . 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 122	. . . . . . . . . . . . . 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 1014	. . . . . . . . . . . . 13 ⊢ (𝑔:𝐴–1-1-onto→ℕ0 → ∀𝑥 ∈ ℕ0 ∀𝑦 ∈ ℕ0 ((◡𝑔‘𝑥) = (◡𝑔‘𝑦) → 𝑥 = 𝑦))
46	45	ad2antrr 488	. . . . . . . . . . . 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 276	. . . . . . . . . . . . 13 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → (𝑛 + 1) ∈ ℕ0)
48		fveqeq2 5608	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑛 + 1) → ((◡𝑔‘𝑥) = (◡𝑔‘𝑦) ↔ (◡𝑔‘(𝑛 + 1)) = (◡𝑔‘𝑦)))
49		eqeq1 2214	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 = 𝑦 ↔ (𝑛 + 1) = 𝑦))
50	48, 49	imbi12d 234	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑛 + 1) → (((◡𝑔‘𝑥) = (◡𝑔‘𝑦) → 𝑥 = 𝑦) ↔ ((◡𝑔‘(𝑛 + 1)) = (◡𝑔‘𝑦) → (𝑛 + 1) = 𝑦)))
51		fveq2 5599	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑗 → (◡𝑔‘𝑦) = (◡𝑔‘𝑗))
52	51	eqeq2d 2219	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑗 → ((◡𝑔‘(𝑛 + 1)) = (◡𝑔‘𝑦) ↔ (◡𝑔‘(𝑛 + 1)) = (◡𝑔‘𝑗)))
53		eqeq2 2217	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑗 → ((𝑛 + 1) = 𝑦 ↔ (𝑛 + 1) = 𝑗))
54	52, 53	imbi12d 234	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑗 → (((◡𝑔‘(𝑛 + 1)) = (◡𝑔‘𝑦) → (𝑛 + 1) = 𝑦) ↔ ((◡𝑔‘(𝑛 + 1)) = (◡𝑔‘𝑗) → (𝑛 + 1) = 𝑗)))
55	50, 54	rspc2v 2897	. . . . . . . . . . . . 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 411	. . . . . . . . . . . 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 665	. . . . . . . . . 10 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → ¬ (◡𝑔‘(𝑛 + 1)) = (◡𝑔‘𝑗))
59	58	neqned 2385	. . . . . . . . 9 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → (◡𝑔‘(𝑛 + 1)) ≠ (◡𝑔‘𝑗))
60	59	ralrimiva 2581	. . . . . . . 8 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) → ∀𝑗 ∈ (0...𝑛)(◡𝑔‘(𝑛 + 1)) ≠ (◡𝑔‘𝑗))
61		fveq2 5599	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑛 + 1) → (◡𝑔‘𝑘) = (◡𝑔‘(𝑛 + 1)))
62	61	neeq1d 2396	. . . . . . . . . 10 ⊢ (𝑘 = (𝑛 + 1) → ((◡𝑔‘𝑘) ≠ (◡𝑔‘𝑗) ↔ (◡𝑔‘(𝑛 + 1)) ≠ (◡𝑔‘𝑗)))
63	62	ralbidv 2508	. . . . . . . . 9 ⊢ (𝑘 = (𝑛 + 1) → (∀𝑗 ∈ (0...𝑛)(◡𝑔‘𝑘) ≠ (◡𝑔‘𝑗) ↔ ∀𝑗 ∈ (0...𝑛)(◡𝑔‘(𝑛 + 1)) ≠ (◡𝑔‘𝑗)))
64	63	rspcev 2884	. . . . . . . 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 411	. . . . . . 7 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) → ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(◡𝑔‘𝑘) ≠ (◡𝑔‘𝑗))
66	65	ralrimiva 2581	. . . . . 6 ⊢ (𝑔:𝐴–1-1-onto→ℕ0 → ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(◡𝑔‘𝑘) ≠ (◡𝑔‘𝑗))
67		cnvexg 5239	. . . . . . . 8 ⊢ (𝑔 ∈ V → ◡𝑔 ∈ V)
68	67	elv 2780	. . . . . . 7 ⊢ ◡𝑔 ∈ V
69		foeq1 5516	. . . . . . . 8 ⊢ (𝑓 = ◡𝑔 → (𝑓:ℕ0–onto→𝐴 ↔ ◡𝑔:ℕ0–onto→𝐴))
70		fveq1 5598	. . . . . . . . . . 11 ⊢ (𝑓 = ◡𝑔 → (𝑓‘𝑘) = (◡𝑔‘𝑘))
71		fveq1 5598	. . . . . . . . . . 11 ⊢ (𝑓 = ◡𝑔 → (𝑓‘𝑗) = (◡𝑔‘𝑗))
72	70, 71	neeq12d 2398	. . . . . . . . . 10 ⊢ (𝑓 = ◡𝑔 → ((𝑓‘𝑘) ≠ (𝑓‘𝑗) ↔ (◡𝑔‘𝑘) ≠ (◡𝑔‘𝑗)))
73	72	rexralbidv 2534	. . . . . . . . 9 ⊢ (𝑓 = ◡𝑔 → (∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝑓‘𝑘) ≠ (𝑓‘𝑗) ↔ ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(◡𝑔‘𝑘) ≠ (◡𝑔‘𝑗)))
74	73	ralbidv 2508	. . . . . . . 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 473	. . . . . . 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 2875	. . . . . 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 411	. . . . 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 306	. . . 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 277	. . 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 1923	. 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 104   ↔ wb 105  DECID wdc 836   ∧ w3a 981   = wceq 1373  ∃wex 1516   ∈ wcel 2178   ≠ wne 2378  ∀wral 2486  ∃wrex 2487  Vcvv 2776   class class class wbr 4059  ^◡ccnv 4692  ran crn 4694   Fn wfn 5285  ⟶wf 5286  –onto→wfo 5288  –1-1-onto→wf1o 5289  ‘cfv 5290  (class class class)co 5967   ≈ cen 6848  0cc0 7960  1c1 7961   + caddc 7963   < clt 8142   ≤ cle 8143  ℕcn 9071  ℕ₀cn0 9330  ℤcz 9407  ...cfz 10165

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-er 6643  df-en 6851  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-n0 9331  df-z 9408  df-uz 9684  df-fz 10166

This theorem is referenced by:  ennnfone  12911