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Theorem eengtrkg 29446

Description: The geometry structure for 𝔼↑𝑁 is a Tarski geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)

**Assertion**
Ref	Expression
eengtrkg	⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG)

Proof of Theorem eengtrkg Dummy variables 𝑎 𝑏 𝑐 𝑓 𝑖 𝑝 𝑠 𝑡 𝑢 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6894	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ V)
2		simpl 488	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑁 ∈ ℕ)
3		simprl 783	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
4		eengbas 29441	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
5	4	adantr 486	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
6	3, 5	eleqtrrd 2863	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (𝔼‘𝑁))
7		simprr 785	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
8	7, 5	eleqtrrd 2863	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (𝔼‘𝑁))
9		axcgrrflx 29374	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁)) → ⟨𝑥, 𝑦⟩Cgr⟨𝑦, 𝑥⟩)
10	2, 6, 8, 9	syl3anc 1398	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → ⟨𝑥, 𝑦⟩Cgr⟨𝑦, 𝑥⟩)
11		eqid 2760	. . . . . . . . 9 ⊢ (Base‘(EEG‘𝑁)) = (Base‘(EEG‘𝑁))
12		eqid 2760	. . . . . . . . 9 ⊢ (dist‘(EEG‘𝑁)) = (dist‘(EEG‘𝑁))
13	2, 11, 12, 3, 7, 7, 3	ecgrtg 29443	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (⟨𝑥, 𝑦⟩Cgr⟨𝑦, 𝑥⟩ ↔ (𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑦(dist‘(EEG‘𝑁))𝑥)))
14	10, 13	mpbid 235	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑦(dist‘(EEG‘𝑁))𝑥))
15	14	ralrimivva 3205	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))(𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑦(dist‘(EEG‘𝑁))𝑥))
16		simpl 488	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑁 ∈ ℕ)
17		simpr1 1213	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
18		simpr2 1214	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
19		simpr3 1215	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑧 ∈ (Base‘(EEG‘𝑁)))
20	16, 11, 12, 17, 18, 19, 19	ecgrtg 29443	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → (⟨𝑥, 𝑦⟩Cgr⟨𝑧, 𝑧⟩ ↔ (𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑧(dist‘(EEG‘𝑁))𝑧)))
21	6	3adantr3 1190	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (𝔼‘𝑁))
22	8	3adantr3 1190	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (𝔼‘𝑁))
23	4	adantr 486	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
24	19, 23	eleqtrrd 2863	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑧 ∈ (𝔼‘𝑁))
25		axcgrid 29376	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁) ∧ 𝑧 ∈ (𝔼‘𝑁))) → (⟨𝑥, 𝑦⟩Cgr⟨𝑧, 𝑧⟩ → 𝑥 = 𝑦))
26	16, 21, 22, 24, 25	syl13anc 1399	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → (⟨𝑥, 𝑦⟩Cgr⟨𝑧, 𝑧⟩ → 𝑥 = 𝑦))
27	20, 26	sylbird 263	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → ((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑧(dist‘(EEG‘𝑁))𝑧) → 𝑥 = 𝑦))
28	27	ralrimivvva 3208	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑧 ∈ (Base‘(EEG‘𝑁))((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑧(dist‘(EEG‘𝑁))𝑧) → 𝑥 = 𝑦))
29	1, 15, 28	jca32 525	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((EEG‘𝑁) ∈ V ∧ (∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))(𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑦(dist‘(EEG‘𝑁))𝑥) ∧ ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑧 ∈ (Base‘(EEG‘𝑁))((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑧(dist‘(EEG‘𝑁))𝑧) → 𝑥 = 𝑦))))
30		eqid 2760	. . . . . 6 ⊢ (Itv‘(EEG‘𝑁)) = (Itv‘(EEG‘𝑁))
31	11, 12, 30	istrkgc 28798	. . . . 5 ⊢ ((EEG‘𝑁) ∈ TarskiG_C ↔ ((EEG‘𝑁) ∈ V ∧ (∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))(𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑦(dist‘(EEG‘𝑁))𝑥) ∧ ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑧 ∈ (Base‘(EEG‘𝑁))((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑧(dist‘(EEG‘𝑁))𝑧) → 𝑥 = 𝑦))))
32	29, 31	sylibr 237	. . . 4 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG_C)
33	2, 11, 30, 3, 3, 7	ebtwntg 29442	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (𝑦 Btwn ⟨𝑥, 𝑥⟩ ↔ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑥)))
34		axbtwnid 29399	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑦 Btwn ⟨𝑥, 𝑥⟩ → 𝑦 = 𝑥))
35	2, 8, 6, 34	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (𝑦 Btwn ⟨𝑥, 𝑥⟩ → 𝑦 = 𝑥))
36	33, 35	sylbird 263	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑥) → 𝑦 = 𝑥))
37	36	imp 412	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑥)) → 𝑦 = 𝑥)
38	37	equcomd 2052	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑥)) → 𝑥 = 𝑦)
39	38	ex 418	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑥) → 𝑥 = 𝑦))
40	39	ralrimivva 3205	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑥) → 𝑥 = 𝑦))
41		simpll 779	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑁 ∈ ℕ)
42	6	adantr 486	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (𝔼‘𝑁))
43	8	adantr 486	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (𝔼‘𝑁))
44	3	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
45	7	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
46		simpr1 1213	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑧 ∈ (Base‘(EEG‘𝑁)))
47	41, 44, 45, 46, 24	syl13anc 1399	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑧 ∈ (𝔼‘𝑁))
48		simpr2 1214	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑢 ∈ (Base‘(EEG‘𝑁)))
49	41, 4	syl 18	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
50	48, 49	eleqtrrd 2863	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑢 ∈ (𝔼‘𝑁))
51		simpr3 1215	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑣 ∈ (Base‘(EEG‘𝑁)))
52	51, 49	eleqtrrd 2863	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑣 ∈ (𝔼‘𝑁))
53		axpasch 29401	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁) ∧ 𝑧 ∈ (𝔼‘𝑁)) ∧ (𝑢 ∈ (𝔼‘𝑁) ∧ 𝑣 ∈ (𝔼‘𝑁))) → ((𝑢 Btwn ⟨𝑥, 𝑧⟩ ∧ 𝑣 Btwn ⟨𝑦, 𝑧⟩) → ∃𝑎 ∈ (𝔼‘𝑁)(𝑎 Btwn ⟨𝑢, 𝑦⟩ ∧ 𝑎 Btwn ⟨𝑣, 𝑥⟩)))
54	41, 42, 43, 47, 50, 52, 53	syl132anc 1415	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → ((𝑢 Btwn ⟨𝑥, 𝑧⟩ ∧ 𝑣 Btwn ⟨𝑦, 𝑧⟩) → ∃𝑎 ∈ (𝔼‘𝑁)(𝑎 Btwn ⟨𝑢, 𝑦⟩ ∧ 𝑎 Btwn ⟨𝑣, 𝑥⟩)))
55	41, 11, 30, 44, 46, 48	ebtwntg 29442	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (𝑢 Btwn ⟨𝑥, 𝑧⟩ ↔ 𝑢 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧)))
56	41, 11, 30, 45, 46, 51	ebtwntg 29442	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (𝑣 Btwn ⟨𝑦, 𝑧⟩ ↔ 𝑣 ∈ (𝑦(Itv‘(EEG‘𝑁))𝑧)))
57	55, 56	anbi12d 644	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → ((𝑢 Btwn ⟨𝑥, 𝑧⟩ ∧ 𝑣 Btwn ⟨𝑦, 𝑧⟩) ↔ (𝑢 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑣 ∈ (𝑦(Itv‘(EEG‘𝑁))𝑧))))
58		simplll 787	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
59	48	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → 𝑢 ∈ (Base‘(EEG‘𝑁)))
60	45	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
61		simpr 490	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → 𝑎 ∈ (𝔼‘𝑁))
62	49	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
63	61, 62	eleqtrd 2862	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → 𝑎 ∈ (Base‘(EEG‘𝑁)))
64	58, 11, 30, 59, 60, 63	ebtwntg 29442	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → (𝑎 Btwn ⟨𝑢, 𝑦⟩ ↔ 𝑎 ∈ (𝑢(Itv‘(EEG‘𝑁))𝑦)))
65	51	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → 𝑣 ∈ (Base‘(EEG‘𝑁)))
66	44	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
67	58, 11, 30, 65, 66, 63	ebtwntg 29442	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → (𝑎 Btwn ⟨𝑣, 𝑥⟩ ↔ 𝑎 ∈ (𝑣(Itv‘(EEG‘𝑁))𝑥)))
68	64, 67	anbi12d 644	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → ((𝑎 Btwn ⟨𝑢, 𝑦⟩ ∧ 𝑎 Btwn ⟨𝑣, 𝑥⟩) ↔ (𝑎 ∈ (𝑢(Itv‘(EEG‘𝑁))𝑦) ∧ 𝑎 ∈ (𝑣(Itv‘(EEG‘𝑁))𝑥))))
69	49, 68	rexeqbidva 3326	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (∃𝑎 ∈ (𝔼‘𝑁)(𝑎 Btwn ⟨𝑢, 𝑦⟩ ∧ 𝑎 Btwn ⟨𝑣, 𝑥⟩) ↔ ∃𝑎 ∈ (Base‘(EEG‘𝑁))(𝑎 ∈ (𝑢(Itv‘(EEG‘𝑁))𝑦) ∧ 𝑎 ∈ (𝑣(Itv‘(EEG‘𝑁))𝑥))))
70	54, 57, 69	3imtr3d 296	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → ((𝑢 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑣 ∈ (𝑦(Itv‘(EEG‘𝑁))𝑧)) → ∃𝑎 ∈ (Base‘(EEG‘𝑁))(𝑎 ∈ (𝑢(Itv‘(EEG‘𝑁))𝑦) ∧ 𝑎 ∈ (𝑣(Itv‘(EEG‘𝑁))𝑥))))
71	70	ralrimivvva 3208	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → ∀𝑧 ∈ (Base‘(EEG‘𝑁))∀𝑢 ∈ (Base‘(EEG‘𝑁))∀𝑣 ∈ (Base‘(EEG‘𝑁))((𝑢 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑣 ∈ (𝑦(Itv‘(EEG‘𝑁))𝑧)) → ∃𝑎 ∈ (Base‘(EEG‘𝑁))(𝑎 ∈ (𝑢(Itv‘(EEG‘𝑁))𝑦) ∧ 𝑎 ∈ (𝑣(Itv‘(EEG‘𝑁))𝑥))))
72	71	ralrimivva 3205	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑧 ∈ (Base‘(EEG‘𝑁))∀𝑢 ∈ (Base‘(EEG‘𝑁))∀𝑣 ∈ (Base‘(EEG‘𝑁))((𝑢 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑣 ∈ (𝑦(Itv‘(EEG‘𝑁))𝑧)) → ∃𝑎 ∈ (Base‘(EEG‘𝑁))(𝑎 ∈ (𝑢(Itv‘(EEG‘𝑁))𝑦) ∧ 𝑎 ∈ (𝑣(Itv‘(EEG‘𝑁))𝑥))))
73		simpl 488	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → 𝑁 ∈ ℕ)
74		elpwi 4564	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) → 𝑠 ⊆ (Base‘(EEG‘𝑁)))
75	74	ad2antrl 741	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → 𝑠 ⊆ (Base‘(EEG‘𝑁)))
76	4	adantr 486	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
77	75, 76	sseqtrrd 3968	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → 𝑠 ⊆ (𝔼‘𝑁))
78		elpwi 4564	. . . . . . . . . . 11 ⊢ (𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)) → 𝑡 ⊆ (Base‘(EEG‘𝑁)))
79	78	ad2antll 742	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → 𝑡 ⊆ (Base‘(EEG‘𝑁)))
80	79, 76	sseqtrrd 3968	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → 𝑡 ⊆ (𝔼‘𝑁))
81		simpll 779	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑠 ⊆ (𝔼‘𝑁) ∧ 𝑡 ⊆ (𝔼‘𝑁))) ∧ ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩) → 𝑁 ∈ ℕ)
82		simplrl 789	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑠 ⊆ (𝔼‘𝑁) ∧ 𝑡 ⊆ (𝔼‘𝑁))) ∧ ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩) → 𝑠 ⊆ (𝔼‘𝑁))
83		simplrr 790	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑠 ⊆ (𝔼‘𝑁) ∧ 𝑡 ⊆ (𝔼‘𝑁))) ∧ ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩) → 𝑡 ⊆ (𝔼‘𝑁))
84		simpr 490	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑠 ⊆ (𝔼‘𝑁) ∧ 𝑡 ⊆ (𝔼‘𝑁))) ∧ ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩) → ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩)
85		axcont 29436	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ⊆ (𝔼‘𝑁) ∧ 𝑡 ⊆ (𝔼‘𝑁) ∧ ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩)) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 Btwn ⟨𝑥, 𝑦⟩)
86	81, 82, 83, 84, 85	syl13anc 1399	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑠 ⊆ (𝔼‘𝑁) ∧ 𝑡 ⊆ (𝔼‘𝑁))) ∧ ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 Btwn ⟨𝑥, 𝑦⟩)
87	86	ex 418	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ⊆ (𝔼‘𝑁) ∧ 𝑡 ⊆ (𝔼‘𝑁))) → (∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩ → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 Btwn ⟨𝑥, 𝑦⟩))
88	73, 77, 80, 87	syl12anc 850	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → (∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩ → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 Btwn ⟨𝑥, 𝑦⟩))
89		simplll 787	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑁 ∈ ℕ)
90		simplr 781	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑎 ∈ (𝔼‘𝑁))
91	76	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
92	90, 91	eleqtrd 2862	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑎 ∈ (Base‘(EEG‘𝑁)))
93	79	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑡 ⊆ (Base‘(EEG‘𝑁)))
94		simprr 785	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑦 ∈ 𝑡)
95	93, 94	sseldd 3932	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
96	75	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑠 ⊆ (Base‘(EEG‘𝑁)))
97		simprl 783	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑥 ∈ 𝑠)
98	96, 97	sseldd 3932	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
99	89, 11, 30, 92, 95, 98	ebtwntg 29442	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → (𝑥 Btwn ⟨𝑎, 𝑦⟩ ↔ 𝑥 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑦)))
100	99	2ralbidva 3224	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → (∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩ ↔ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑦)))
101	76, 100	rexeqbidva 3326	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → (∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩ ↔ ∃𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑦)))
102		simplll 787	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑁 ∈ ℕ)
103	75	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑠 ⊆ (Base‘(EEG‘𝑁)))
104		simprl 783	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑥 ∈ 𝑠)
105	103, 104	sseldd 3932	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
106	79	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑡 ⊆ (Base‘(EEG‘𝑁)))
107		simprr 785	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑦 ∈ 𝑡)
108	106, 107	sseldd 3932	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
109		simplr 781	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑏 ∈ (𝔼‘𝑁))
110	76	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
111	109, 110	eleqtrd 2862	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑏 ∈ (Base‘(EEG‘𝑁)))
112	102, 11, 30, 105, 108, 111	ebtwntg 29442	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → (𝑏 Btwn ⟨𝑥, 𝑦⟩ ↔ 𝑏 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦)))
113	112	2ralbidva 3224	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) → (∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 Btwn ⟨𝑥, 𝑦⟩ ↔ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦)))
114	76, 113	rexeqbidva 3326	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → (∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 Btwn ⟨𝑥, 𝑦⟩ ↔ ∃𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦)))
115	88, 101, 114	3imtr3d 296	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → (∃𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑦) → ∃𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦)))
116	115	ralrimivva 3205	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ∀𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁))∀𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁))(∃𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑦) → ∃𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦)))
117	40, 72, 116	3jca 1146	. . . . 5 ⊢ (𝑁 ∈ ℕ → (∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑧 ∈ (Base‘(EEG‘𝑁))∀𝑢 ∈ (Base‘(EEG‘𝑁))∀𝑣 ∈ (Base‘(EEG‘𝑁))((𝑢 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑣 ∈ (𝑦(Itv‘(EEG‘𝑁))𝑧)) → ∃𝑎 ∈ (Base‘(EEG‘𝑁))(𝑎 ∈ (𝑢(Itv‘(EEG‘𝑁))𝑦) ∧ 𝑎 ∈ (𝑣(Itv‘(EEG‘𝑁))𝑥))) ∧ ∀𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁))∀𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁))(∃𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑦) → ∃𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦))))
118	11, 12, 30	istrkgb 28799	. . . . 5 ⊢ ((EEG‘𝑁) ∈ TarskiG_B ↔ ((EEG‘𝑁) ∈ V ∧ (∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑧 ∈ (Base‘(EEG‘𝑁))∀𝑢 ∈ (Base‘(EEG‘𝑁))∀𝑣 ∈ (Base‘(EEG‘𝑁))((𝑢 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑣 ∈ (𝑦(Itv‘(EEG‘𝑁))𝑧)) → ∃𝑎 ∈ (Base‘(EEG‘𝑁))(𝑎 ∈ (𝑢(Itv‘(EEG‘𝑁))𝑦) ∧ 𝑎 ∈ (𝑣(Itv‘(EEG‘𝑁))𝑥))) ∧ ∀𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁))∀𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁))(∃𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑦) → ∃𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦)))))
119	1, 117, 118	sylanbrc 595	. . . 4 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG_B)
120	32, 119	elind 4146	. . 3 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ (TarskiG_C ∩ TarskiG_B))
121		simplll 787	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑁 ∈ ℕ)
122	3	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
123	121, 4	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
124	122, 123	eleqtrrd 2863	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (𝔼‘𝑁))
125	7	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
126	125, 123	eleqtrrd 2863	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (𝔼‘𝑁))
127		simplr1 1234	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑧 ∈ (Base‘(EEG‘𝑁)))
128	127, 123	eleqtrrd 2863	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑧 ∈ (𝔼‘𝑁))
129		simplr2 1235	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑢 ∈ (Base‘(EEG‘𝑁)))
130	129, 123	eleqtrrd 2863	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑢 ∈ (𝔼‘𝑁))
131		simplr3 1236	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑎 ∈ (Base‘(EEG‘𝑁)))
132	131, 123	eleqtrrd 2863	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑎 ∈ (𝔼‘𝑁))
133		simpr1 1213	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑏 ∈ (Base‘(EEG‘𝑁)))
134	133, 123	eleqtrrd 2863	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑏 ∈ (𝔼‘𝑁))
135		simpr2 1214	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑐 ∈ (Base‘(EEG‘𝑁)))
136	135, 123	eleqtrrd 2863	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑐 ∈ (𝔼‘𝑁))
137		simpr3 1215	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑣 ∈ (Base‘(EEG‘𝑁)))
138	137, 123	eleqtrrd 2863	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑣 ∈ (𝔼‘𝑁))
139		3anass 1111	. . . . . . . . . . . 12 ⊢ (((𝑥 ≠ 𝑦 ∧ 𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ 𝑏 Btwn ⟨𝑎, 𝑐⟩) ∧ (⟨𝑥, 𝑦⟩Cgr⟨𝑎, 𝑏⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑏, 𝑐⟩) ∧ (⟨𝑥, 𝑢⟩Cgr⟨𝑎, 𝑣⟩ ∧ ⟨𝑦, 𝑢⟩Cgr⟨𝑏, 𝑣⟩)) ↔ ((𝑥 ≠ 𝑦 ∧ 𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ 𝑏 Btwn ⟨𝑎, 𝑐⟩) ∧ ((⟨𝑥, 𝑦⟩Cgr⟨𝑎, 𝑏⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑏, 𝑐⟩) ∧ (⟨𝑥, 𝑢⟩Cgr⟨𝑎, 𝑣⟩ ∧ ⟨𝑦, 𝑢⟩Cgr⟨𝑏, 𝑣⟩))))
140		ax5seg 29398	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑧 ∈ (𝔼‘𝑁) ∧ 𝑢 ∈ (𝔼‘𝑁) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑏 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁) ∧ 𝑣 ∈ (𝔼‘𝑁))) → (((𝑥 ≠ 𝑦 ∧ 𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ 𝑏 Btwn ⟨𝑎, 𝑐⟩) ∧ (⟨𝑥, 𝑦⟩Cgr⟨𝑎, 𝑏⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑏, 𝑐⟩) ∧ (⟨𝑥, 𝑢⟩Cgr⟨𝑎, 𝑣⟩ ∧ ⟨𝑦, 𝑢⟩Cgr⟨𝑏, 𝑣⟩)) → ⟨𝑧, 𝑢⟩Cgr⟨𝑐, 𝑣⟩))
141	139, 140	biimtrrid 246	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑧 ∈ (𝔼‘𝑁) ∧ 𝑢 ∈ (𝔼‘𝑁) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑏 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁) ∧ 𝑣 ∈ (𝔼‘𝑁))) → (((𝑥 ≠ 𝑦 ∧ 𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ 𝑏 Btwn ⟨𝑎, 𝑐⟩) ∧ ((⟨𝑥, 𝑦⟩Cgr⟨𝑎, 𝑏⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑏, 𝑐⟩) ∧ (⟨𝑥, 𝑢⟩Cgr⟨𝑎, 𝑣⟩ ∧ ⟨𝑦, 𝑢⟩Cgr⟨𝑏, 𝑣⟩))) → ⟨𝑧, 𝑢⟩Cgr⟨𝑐, 𝑣⟩))
142	121, 124, 126, 128, 130, 132, 134, 136, 138, 141	syl333anc 1429	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (((𝑥 ≠ 𝑦 ∧ 𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ 𝑏 Btwn ⟨𝑎, 𝑐⟩) ∧ ((⟨𝑥, 𝑦⟩Cgr⟨𝑎, 𝑏⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑏, 𝑐⟩) ∧ (⟨𝑥, 𝑢⟩Cgr⟨𝑎, 𝑣⟩ ∧ ⟨𝑦, 𝑢⟩Cgr⟨𝑏, 𝑣⟩))) → ⟨𝑧, 𝑢⟩Cgr⟨𝑐, 𝑣⟩))
143	121, 11, 30, 122, 127, 125	ebtwntg 29442	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (𝑦 Btwn ⟨𝑥, 𝑧⟩ ↔ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧)))
144	121, 11, 30, 131, 135, 133	ebtwntg 29442	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (𝑏 Btwn ⟨𝑎, 𝑐⟩ ↔ 𝑏 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑐)))
145	143, 144	3anbi23d 1467	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → ((𝑥 ≠ 𝑦 ∧ 𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ 𝑏 Btwn ⟨𝑎, 𝑐⟩) ↔ (𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑏 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑐))))
146	121, 11, 12, 122, 125, 131, 133	ecgrtg 29443	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (⟨𝑥, 𝑦⟩Cgr⟨𝑎, 𝑏⟩ ↔ (𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑎(dist‘(EEG‘𝑁))𝑏)))
147	121, 11, 12, 125, 127, 133, 135	ecgrtg 29443	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (⟨𝑦, 𝑧⟩Cgr⟨𝑏, 𝑐⟩ ↔ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑏(dist‘(EEG‘𝑁))𝑐)))
148	146, 147	anbi12d 644	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → ((⟨𝑥, 𝑦⟩Cgr⟨𝑎, 𝑏⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑏, 𝑐⟩) ↔ ((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑎(dist‘(EEG‘𝑁))𝑏) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑏(dist‘(EEG‘𝑁))𝑐))))
149	121, 11, 12, 122, 129, 131, 137	ecgrtg 29443	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (⟨𝑥, 𝑢⟩Cgr⟨𝑎, 𝑣⟩ ↔ (𝑥(dist‘(EEG‘𝑁))𝑢) = (𝑎(dist‘(EEG‘𝑁))𝑣)))
150	121, 11, 12, 125, 129, 133, 137	ecgrtg 29443	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (⟨𝑦, 𝑢⟩Cgr⟨𝑏, 𝑣⟩ ↔ (𝑦(dist‘(EEG‘𝑁))𝑢) = (𝑏(dist‘(EEG‘𝑁))𝑣)))
151	149, 150	anbi12d 644	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → ((⟨𝑥, 𝑢⟩Cgr⟨𝑎, 𝑣⟩ ∧ ⟨𝑦, 𝑢⟩Cgr⟨𝑏, 𝑣⟩) ↔ ((𝑥(dist‘(EEG‘𝑁))𝑢) = (𝑎(dist‘(EEG‘𝑁))𝑣) ∧ (𝑦(dist‘(EEG‘𝑁))𝑢) = (𝑏(dist‘(EEG‘𝑁))𝑣))))
152	148, 151	anbi12d 644	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (((⟨𝑥, 𝑦⟩Cgr⟨𝑎, 𝑏⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑏, 𝑐⟩) ∧ (⟨𝑥, 𝑢⟩Cgr⟨𝑎, 𝑣⟩ ∧ ⟨𝑦, 𝑢⟩Cgr⟨𝑏, 𝑣⟩)) ↔ (((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑎(dist‘(EEG‘𝑁))𝑏) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑏(dist‘(EEG‘𝑁))𝑐)) ∧ ((𝑥(dist‘(EEG‘𝑁))𝑢) = (𝑎(dist‘(EEG‘𝑁))𝑣) ∧ (𝑦(dist‘(EEG‘𝑁))𝑢) = (𝑏(dist‘(EEG‘𝑁))𝑣)))))
153	145, 152	anbi12d 644	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (((𝑥 ≠ 𝑦 ∧ 𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ 𝑏 Btwn ⟨𝑎, 𝑐⟩) ∧ ((⟨𝑥, 𝑦⟩Cgr⟨𝑎, 𝑏⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑏, 𝑐⟩) ∧ (⟨𝑥, 𝑢⟩Cgr⟨𝑎, 𝑣⟩ ∧ ⟨𝑦, 𝑢⟩Cgr⟨𝑏, 𝑣⟩))) ↔ ((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑏 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑐)) ∧ (((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑎(dist‘(EEG‘𝑁))𝑏) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑏(dist‘(EEG‘𝑁))𝑐)) ∧ ((𝑥(dist‘(EEG‘𝑁))𝑢) = (𝑎(dist‘(EEG‘𝑁))𝑣) ∧ (𝑦(dist‘(EEG‘𝑁))𝑢) = (𝑏(dist‘(EEG‘𝑁))𝑣))))))
154	121, 11, 12, 127, 129, 135, 137	ecgrtg 29443	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (⟨𝑧, 𝑢⟩Cgr⟨𝑐, 𝑣⟩ ↔ (𝑧(dist‘(EEG‘𝑁))𝑢) = (𝑐(dist‘(EEG‘𝑁))𝑣)))
155	142, 153, 154	3imtr3d 296	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑏 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑐)) ∧ (((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑎(dist‘(EEG‘𝑁))𝑏) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑏(dist‘(EEG‘𝑁))𝑐)) ∧ ((𝑥(dist‘(EEG‘𝑁))𝑢) = (𝑎(dist‘(EEG‘𝑁))𝑣) ∧ (𝑦(dist‘(EEG‘𝑁))𝑢) = (𝑏(dist‘(EEG‘𝑁))𝑣)))) → (𝑧(dist‘(EEG‘𝑁))𝑢) = (𝑐(dist‘(EEG‘𝑁))𝑣)))
156	155	ralrimivvva 3208	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) → ∀𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑐 ∈ (Base‘(EEG‘𝑁))∀𝑣 ∈ (Base‘(EEG‘𝑁))(((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑏 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑐)) ∧ (((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑎(dist‘(EEG‘𝑁))𝑏) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑏(dist‘(EEG‘𝑁))𝑐)) ∧ ((𝑥(dist‘(EEG‘𝑁))𝑢) = (𝑎(dist‘(EEG‘𝑁))𝑣) ∧ (𝑦(dist‘(EEG‘𝑁))𝑢) = (𝑏(dist‘(EEG‘𝑁))𝑣)))) → (𝑧(dist‘(EEG‘𝑁))𝑢) = (𝑐(dist‘(EEG‘𝑁))𝑣)))
157	156	ralrimivvva 3208	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → ∀𝑧 ∈ (Base‘(EEG‘𝑁))∀𝑢 ∈ (Base‘(EEG‘𝑁))∀𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑐 ∈ (Base‘(EEG‘𝑁))∀𝑣 ∈ (Base‘(EEG‘𝑁))(((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑏 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑐)) ∧ (((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑎(dist‘(EEG‘𝑁))𝑏) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑏(dist‘(EEG‘𝑁))𝑐)) ∧ ((𝑥(dist‘(EEG‘𝑁))𝑢) = (𝑎(dist‘(EEG‘𝑁))𝑣) ∧ (𝑦(dist‘(EEG‘𝑁))𝑢) = (𝑏(dist‘(EEG‘𝑁))𝑣)))) → (𝑧(dist‘(EEG‘𝑁))𝑢) = (𝑐(dist‘(EEG‘𝑁))𝑣)))
158	157	ralrimivva 3205	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑧 ∈ (Base‘(EEG‘𝑁))∀𝑢 ∈ (Base‘(EEG‘𝑁))∀𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑐 ∈ (Base‘(EEG‘𝑁))∀𝑣 ∈ (Base‘(EEG‘𝑁))(((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑏 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑐)) ∧ (((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑎(dist‘(EEG‘𝑁))𝑏) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑏(dist‘(EEG‘𝑁))𝑐)) ∧ ((𝑥(dist‘(EEG‘𝑁))𝑢) = (𝑎(dist‘(EEG‘𝑁))𝑣) ∧ (𝑦(dist‘(EEG‘𝑁))𝑢) = (𝑏(dist‘(EEG‘𝑁))𝑣)))) → (𝑧(dist‘(EEG‘𝑁))𝑢) = (𝑐(dist‘(EEG‘𝑁))𝑣)))
159		simpll 779	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → 𝑁 ∈ ℕ)
160	6	adantr 486	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (𝔼‘𝑁))
161	8	adantr 486	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (𝔼‘𝑁))
162		simprl 783	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → 𝑎 ∈ (Base‘(EEG‘𝑁)))
163	159, 4	syl 18	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
164	162, 163	eleqtrrd 2863	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → 𝑎 ∈ (𝔼‘𝑁))
165		simprr 785	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → 𝑏 ∈ (Base‘(EEG‘𝑁)))
166	165, 163	eleqtrrd 2863	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → 𝑏 ∈ (𝔼‘𝑁))
167		axsegcon 29387	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑎 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁))) → ∃𝑧 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑎, 𝑏⟩))
168	159, 160, 161, 164, 166, 167	syl122anc 1406	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → ∃𝑧 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑎, 𝑏⟩))
169		simplll 787	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
170	3	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
171		simpr 490	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → 𝑧 ∈ (𝔼‘𝑁))
172	163	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
173	171, 172	eleqtrd 2862	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → 𝑧 ∈ (Base‘(EEG‘𝑁)))
174	7	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
175	169, 11, 30, 170, 173, 174	ebtwntg 29442	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → (𝑦 Btwn ⟨𝑥, 𝑧⟩ ↔ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧)))
176		simplrl 789	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → 𝑎 ∈ (Base‘(EEG‘𝑁)))
177		simplrr 790	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → 𝑏 ∈ (Base‘(EEG‘𝑁)))
178	169, 11, 12, 174, 173, 176, 177	ecgrtg 29443	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → (⟨𝑦, 𝑧⟩Cgr⟨𝑎, 𝑏⟩ ↔ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑎(dist‘(EEG‘𝑁))𝑏)))
179	175, 178	anbi12d 644	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → ((𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑎, 𝑏⟩) ↔ (𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑎(dist‘(EEG‘𝑁))𝑏))))
180	163, 179	rexeqbidva 3326	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → (∃𝑧 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑎, 𝑏⟩) ↔ ∃𝑧 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑎(dist‘(EEG‘𝑁))𝑏))))
181	168, 180	mpbid 235	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → ∃𝑧 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑎(dist‘(EEG‘𝑁))𝑏)))
182	181	ralrimivva 3205	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → ∀𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑏 ∈ (Base‘(EEG‘𝑁))∃𝑧 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑎(dist‘(EEG‘𝑁))𝑏)))
183	182	ralrimivva 3205	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑏 ∈ (Base‘(EEG‘𝑁))∃𝑧 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑎(dist‘(EEG‘𝑁))𝑏)))
184	1, 158, 183	jca32 525	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((EEG‘𝑁) ∈ V ∧ (∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑧 ∈ (Base‘(EEG‘𝑁))∀𝑢 ∈ (Base‘(EEG‘𝑁))∀𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑐 ∈ (Base‘(EEG‘𝑁))∀𝑣 ∈ (Base‘(EEG‘𝑁))(((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑏 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑐)) ∧ (((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑎(dist‘(EEG‘𝑁))𝑏) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑏(dist‘(EEG‘𝑁))𝑐)) ∧ ((𝑥(dist‘(EEG‘𝑁))𝑢) = (𝑎(dist‘(EEG‘𝑁))𝑣) ∧ (𝑦(dist‘(EEG‘𝑁))𝑢) = (𝑏(dist‘(EEG‘𝑁))𝑣)))) → (𝑧(dist‘(EEG‘𝑁))𝑢) = (𝑐(dist‘(EEG‘𝑁))𝑣)) ∧ ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑏 ∈ (Base‘(EEG‘𝑁))∃𝑧 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑎(dist‘(EEG‘𝑁))𝑏)))))
185	11, 12, 30	istrkgcb 28800	. . . . 5 ⊢ ((EEG‘𝑁) ∈ TarskiG_CB ↔ ((EEG‘𝑁) ∈ V ∧ (∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑧 ∈ (Base‘(EEG‘𝑁))∀𝑢 ∈ (Base‘(EEG‘𝑁))∀𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑐 ∈ (Base‘(EEG‘𝑁))∀𝑣 ∈ (Base‘(EEG‘𝑁))(((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑏 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑐)) ∧ (((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑎(dist‘(EEG‘𝑁))𝑏) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑏(dist‘(EEG‘𝑁))𝑐)) ∧ ((𝑥(dist‘(EEG‘𝑁))𝑢) = (𝑎(dist‘(EEG‘𝑁))𝑣) ∧ (𝑦(dist‘(EEG‘𝑁))𝑢) = (𝑏(dist‘(EEG‘𝑁))𝑣)))) → (𝑧(dist‘(EEG‘𝑁))𝑢) = (𝑐(dist‘(EEG‘𝑁))𝑣)) ∧ ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑏 ∈ (Base‘(EEG‘𝑁))∃𝑧 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑎(dist‘(EEG‘𝑁))𝑏)))))
186	184, 185	sylibr 237	. . . 4 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG_CB)
187	11, 30	elntg 29444	. . . . 5 ⊢ (𝑁 ∈ ℕ → (LineG‘(EEG‘𝑁)) = (𝑥 ∈ (Base‘(EEG‘𝑁)), 𝑦 ∈ ((Base‘(EEG‘𝑁)) ∖ {𝑥}) ↦ {𝑧 ∈ (Base‘(EEG‘𝑁)) ∣ (𝑧 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑥 ∈ (𝑧(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧))}))
188	11, 12, 30	istrkgl 28802	. . . . 5 ⊢ ((EEG‘𝑁) ∈ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})} ↔ ((EEG‘𝑁) ∈ V ∧ (LineG‘(EEG‘𝑁)) = (𝑥 ∈ (Base‘(EEG‘𝑁)), 𝑦 ∈ ((Base‘(EEG‘𝑁)) ∖ {𝑥}) ↦ {𝑧 ∈ (Base‘(EEG‘𝑁)) ∣ (𝑧 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑥 ∈ (𝑧(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧))})))
189	1, 187, 188	sylanbrc 595	. . . 4 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})
190	186, 189	elind 4146	. . 3 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ (TarskiG_CB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
191	120, 190	elind 4146	. 2 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ ((TarskiG_C ∩ TarskiG_B) ∩ (TarskiG_CB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})))
192		df-trkg 28797	. 2 ⊢ TarskiG = ((TarskiG_C ∩ TarskiG_B) ∩ (TarskiG_CB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
193	191, 192	eleqtrrdi 2871	1 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∨ w3o 1102 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 {cab 2738 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 {crab 3412 Vcvv 3450 [wsbc 3739 ∖ cdif 3896 ∩ cin 3898 ⊆ wss 3899 𝒫 cpw 4557 {csn 4584 ⟨cop 4590 class class class wbr 5103 ‘cfv 6533 (class class class)co 7414 ∈ cmpo 7416 ℕcn 12260 Basecbs 17304 distcds 17354 TarskiGcstrkg 28771 TarskiG_Ccstrkgc 28772 TarskiG_Bcstrkgb 28773 TarskiG_CBcstrkgcb 28774 Itvcitv 28777 LineGclng 28778 𝔼cee 29347 Btwn cbtwn 29348 Cgrccgr 29349 EEGceeng 29437

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-inf2 9623 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8458 df-er 8699 df-map 8831 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9415 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-rp 13046 df-ico 13407 df-icc 13408 df-fz 13565 df-fzo 13713 df-seq 14069 df-exp 14129 df-hash 14398 df-cj 15189 df-re 15190 df-im 15191 df-sqrt 15325 df-abs 15326 df-clim 15578 df-sum 15777 df-struct 17242 df-slot 17277 df-ndx 17289 df-base 17305 df-ds 17367 df-itv 28779 df-lng 28780 df-trkgc 28792 df-trkgb 28793 df-trkgcb 28794 df-trkg 28797 df-ee 29350 df-btwn 29351 df-cgr 29352 df-eeng 29438

This theorem is used by: (None)

W3C validator