Theorem ennnfonelemim 13315

Description: Lemma for ennnfone 13316. 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 10870	. . . 4 ⊢ ℕ0 ≈ ℕ
2	1	ensymi 7069	. . 3 ⊢ ℕ ≈ ℕ0
3		entr 7071	. . 3 ⊢ ((𝐴 ≈ ℕ ∧ ℕ ≈ ℕ0) → 𝐴 ≈ ℕ0)
4	2, 3	mpan2 429	. 2 ⊢ (𝐴 ≈ ℕ → 𝐴 ≈ ℕ0)
5		bren 7030	. . . 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 5639	. . . . . . . . . . 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 535	. . . . . . . . . 10 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
10	8, 9	ffvelcdmd 5844	. . . . . . . . 9 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑔‘𝑥) ∈ ℕ0)
11	10	nn0zd 9766	. . . . . . . 8 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑔‘𝑥) ∈ ℤ)
12		simprr 537	. . . . . . . . . 10 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
13	8, 12	ffvelcdmd 5844	. . . . . . . . 9 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑔‘𝑦) ∈ ℕ0)
14	13	nn0zd 9766	. . . . . . . 8 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑔‘𝑦) ∈ ℤ)
15		zdceq 9720	. . . . . . . 8 ⊢ (((𝑔‘𝑥) ∈ ℤ ∧ (𝑔‘𝑦) ∈ ℤ) → DECID (𝑔‘𝑥) = (𝑔‘𝑦))
16	11, 14, 15	syl2anc 415	. . . . . . 7 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → DECID (𝑔‘𝑥) = (𝑔‘𝑦))
17		dff1o6 5982	. . . . . . . . . . . . 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 1045	. . . . . . . . . . . 12 ⊢ (𝑔:𝐴–1-1-onto→ℕ0 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑔‘𝑥) = (𝑔‘𝑦) → 𝑥 = 𝑦))
19	18	r19.21bi 2638	. . . . . . . . . . 11 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 ((𝑔‘𝑥) = (𝑔‘𝑦) → 𝑥 = 𝑦))
20	19	r19.21bi 2638	. . . . . . . . . 10 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝑔‘𝑥) = (𝑔‘𝑦) → 𝑥 = 𝑦))
21	20	anasss 403	. . . . . . . . 9 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑔‘𝑥) = (𝑔‘𝑦) → 𝑥 = 𝑦))
22		fveq2 5695	. . . . . . . . 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 850	. . . . . . 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 2632	. . . . 5 ⊢ (𝑔:𝐴–1-1-onto→ℕ0 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
27		f1ocnv 5652	. . . . . . 7 ⊢ (𝑔:𝐴–1-1-onto→ℕ0 → ◡𝑔:ℕ0–1-1-onto→𝐴)
28		f1ofo 5646	. . . . . . 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 9603	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
31	30	adantl 277	. . . . . . . 8 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) → (𝑛 + 1) ∈ ℕ0)
32		elfznn0 10521	. . . . . . . . . . . . . . 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 9621	. . . . . . . . . . . . 13 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → 𝑗 ∈ ℝ)
35		elfzle2 10432	. . . . . . . . . . . . . . 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 533	. . . . . . . . . . . . . . 15 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → 𝑛 ∈ ℕ0)
38		nn0leltp1 9708	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0) → (𝑗 ≤ 𝑛 ↔ 𝑗 < (𝑛 + 1)))
39	33, 37, 38	syl2anc 415	. . . . . . . . . . . . . 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 8438	. . . . . . . . . . . 12 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → (𝑛 + 1) ≠ 𝑗)
42	41	neneqd 2441	. . . . . . . . . . 11 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → ¬ (𝑛 + 1) = 𝑗)
43		dff1o6 5982	. . . . . . . . . . . . . . 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 1042	. . . . . . . . . . . . 13 ⊢ (𝑔:𝐴–1-1-onto→ℕ0 → ∀𝑥 ∈ ℕ0 ∀𝑦 ∈ ℕ0 ((◡𝑔‘𝑥) = (◡𝑔‘𝑦) → 𝑥 = 𝑦))
46	45	ad2antrr 492	. . . . . . . . . . . 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 5704	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑛 + 1) → ((◡𝑔‘𝑥) = (◡𝑔‘𝑦) ↔ (◡𝑔‘(𝑛 + 1)) = (◡𝑔‘𝑦)))
49		eqeq1 2245	. . . . . . . . . . . . . . 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 5695	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑗 → (◡𝑔‘𝑦) = (◡𝑔‘𝑗))
52	51	eqeq2d 2250	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑗 → ((◡𝑔‘(𝑛 + 1)) = (◡𝑔‘𝑦) ↔ (◡𝑔‘(𝑛 + 1)) = (◡𝑔‘𝑗)))
53		eqeq2 2248	. . . . . . . . . . . . . . 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 2943	. . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 673	. . . . . . . . . 10 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → ¬ (◡𝑔‘(𝑛 + 1)) = (◡𝑔‘𝑗))
59	58	neqned 2427	. . . . . . . . 9 ⊢ (((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑛)) → (◡𝑔‘(𝑛 + 1)) ≠ (◡𝑔‘𝑗))
60	59	ralrimiva 2623	. . . . . . . 8 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) → ∀𝑗 ∈ (0...𝑛)(◡𝑔‘(𝑛 + 1)) ≠ (◡𝑔‘𝑗))
61		fveq2 5695	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑛 + 1) → (◡𝑔‘𝑘) = (◡𝑔‘(𝑛 + 1)))
62	61	neeq1d 2438	. . . . . . . . . 10 ⊢ (𝑘 = (𝑛 + 1) → ((◡𝑔‘𝑘) ≠ (◡𝑔‘𝑗) ↔ (◡𝑔‘(𝑛 + 1)) ≠ (◡𝑔‘𝑗)))
63	62	ralbidv 2550	. . . . . . . . 9 ⊢ (𝑘 = (𝑛 + 1) → (∀𝑗 ∈ (0...𝑛)(◡𝑔‘𝑘) ≠ (◡𝑔‘𝑗) ↔ ∀𝑗 ∈ (0...𝑛)(◡𝑔‘(𝑛 + 1)) ≠ (◡𝑔‘𝑗)))
64	63	rspcev 2929	. . . . . . . 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 415	. . . . . . 7 ⊢ ((𝑔:𝐴–1-1-onto→ℕ0 ∧ 𝑛 ∈ ℕ0) → ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(◡𝑔‘𝑘) ≠ (◡𝑔‘𝑗))
66	65	ralrimiva 2623	. . . . . 6 ⊢ (𝑔:𝐴–1-1-onto→ℕ0 → ∀𝑛 ∈ ℕ0 ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(◡𝑔‘𝑘) ≠ (◡𝑔‘𝑗))
67		cnvexg 5325	. . . . . . . 8 ⊢ (𝑔 ∈ V → ◡𝑔 ∈ V)
68	67	elv 2825	. . . . . . 7 ⊢ ◡𝑔 ∈ V
69		foeq1 5611	. . . . . . . 8 ⊢ (𝑓 = ◡𝑔 → (𝑓:ℕ0–onto→𝐴 ↔ ◡𝑔:ℕ0–onto→𝐴))
70		fveq1 5694	. . . . . . . . . . 11 ⊢ (𝑓 = ◡𝑔 → (𝑓‘𝑘) = (◡𝑔‘𝑘))
71		fveq1 5694	. . . . . . . . . . 11 ⊢ (𝑓 = ◡𝑔 → (𝑓‘𝑗) = (◡𝑔‘𝑗))
72	70, 71	neeq12d 2440	. . . . . . . . . 10 ⊢ (𝑓 = ◡𝑔 → ((𝑓‘𝑘) ≠ (𝑓‘𝑗) ↔ (◡𝑔‘𝑘) ≠ (◡𝑔‘𝑗)))
73	72	rexralbidv 2576	. . . . . . . . 9 ⊢ (𝑓 = ◡𝑔 → (∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(𝑓‘𝑘) ≠ (𝑓‘𝑗) ↔ ∃𝑘 ∈ ℕ0 ∀𝑗 ∈ (0...𝑛)(◡𝑔‘𝑘) ≠ (◡𝑔‘𝑗)))
74	73	ralbidv 2550	. . . . . . . 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 477	. . . . . . 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 2920	. . . . . 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 415	. . . . 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 1954	. 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))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 846   ∧ w3a 1009   = wceq 1402  ∃wex 1545   ∈ wcel 2209   ≠ wne 2420  ∀wral 2528  ∃wrex 2529  Vcvv 2821   class class class wbr 4130  ^◡ccnv 4773  ran crn 4775   Fn wfn 5372  ⟶wf 5373  –onto→wfo 5375  –1-1-onto→wf1o 5376  ‘cfv 5377  (class class class)co 6085   ≈ cen 7020  0cc0 8179  1c1 8180   + caddc 8182   < clt 8360   ≤ cle 8361  ℕcn 9304  ℕ₀cn0 9563  ℤcz 9644  ...cfz 10411

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-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295

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-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  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-er 6807  df-en 7023  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412

This theorem is used by:  ennnfone  13316