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Theorem eengtrkg 28917
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.
StepHypRef Expression
1 fvexd 6908 . . . . . 6 (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ V)
2 simpl 481 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑁 ∈ ℕ)
3 simprl 769 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
4 eengbas 28912 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
54adantr 479 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
63, 5eleqtrrd 2829 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (𝔼‘𝑁))
7 simprr 771 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
87, 5eleqtrrd 2829 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (𝔼‘𝑁))
9 axcgrrflx 28845 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁)) → ⟨𝑥, 𝑦⟩Cgr⟨𝑦, 𝑥⟩)
102, 6, 8, 9syl3anc 1368 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → ⟨𝑥, 𝑦⟩Cgr⟨𝑦, 𝑥⟩)
11 eqid 2726 . . . . . . . . 9 (Base‘(EEG‘𝑁)) = (Base‘(EEG‘𝑁))
12 eqid 2726 . . . . . . . . 9 (dist‘(EEG‘𝑁)) = (dist‘(EEG‘𝑁))
132, 11, 12, 3, 7, 7, 3ecgrtg 28914 . . . . . . . 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)))
1410, 13mpbid 231 . . . . . . 7 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑦(dist‘(EEG‘𝑁))𝑥))
1514ralrimivva 3191 . . . . . 6 (𝑁 ∈ ℕ → ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))(𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑦(dist‘(EEG‘𝑁))𝑥))
16 simpl 481 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑁 ∈ ℕ)
17 simpr1 1191 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
18 simpr2 1192 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
19 simpr3 1193 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑧 ∈ (Base‘(EEG‘𝑁)))
2016, 11, 12, 17, 18, 19, 19ecgrtg 28914 . . . . . . . 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)))
2163adantr3 1168 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (𝔼‘𝑁))
2283adantr3 1168 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (𝔼‘𝑁))
234adantr 479 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
2419, 23eleqtrrd 2829 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑧 ∈ (𝔼‘𝑁))
25 axcgrid 28847 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁) ∧ 𝑧 ∈ (𝔼‘𝑁))) → (⟨𝑥, 𝑦⟩Cgr⟨𝑧, 𝑧⟩ → 𝑥 = 𝑦))
2616, 21, 22, 24, 25syl13anc 1369 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → (⟨𝑥, 𝑦⟩Cgr⟨𝑧, 𝑧⟩ → 𝑥 = 𝑦))
2720, 26sylbird 259 . . . . . . 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))
2827ralrimivvva 3194 . . . . . 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))
291, 15, 28jca32 514 . . . . 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 2726 . . . . . 6 (Itv‘(EEG‘𝑁)) = (Itv‘(EEG‘𝑁))
3111, 12, 30istrkgc 28378 . . . . 5 ((EEG‘𝑁) ∈ TarskiGC ↔ ((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))))
3229, 31sylibr 233 . . . 4 (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiGC)
332, 11, 30, 3, 3, 7ebtwntg 28913 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (𝑦 Btwn ⟨𝑥, 𝑥⟩ ↔ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑥)))
34 axbtwnid 28870 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝑦 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑦 Btwn ⟨𝑥, 𝑥⟩ → 𝑦 = 𝑥))
352, 8, 6, 34syl3anc 1368 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (𝑦 Btwn ⟨𝑥, 𝑥⟩ → 𝑦 = 𝑥))
3633, 35sylbird 259 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑥) → 𝑦 = 𝑥))
3736imp 405 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑥)) → 𝑦 = 𝑥)
3837equcomd 2015 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑥)) → 𝑥 = 𝑦)
3938ex 411 . . . . . . 7 ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑥) → 𝑥 = 𝑦))
4039ralrimivva 3191 . . . . . 6 (𝑁 ∈ ℕ → ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑥) → 𝑥 = 𝑦))
41 simpll 765 . . . . . . . . . 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)
426adantr 479 . . . . . . . . . 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))
438adantr 479 . . . . . . . . . 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))
443adantr 479 . . . . . . . . . . 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)))
457adantr 479 . . . . . . . . . . 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 1191 . . . . . . . . . . 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)))
4741, 44, 45, 46, 24syl13anc 1369 . . . . . . . . . 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 1192 . . . . . . . . . . 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)))
4941, 4syl 17 . . . . . . . . . . 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)))
5048, 49eleqtrrd 2829 . . . . . . . . . 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 1193 . . . . . . . . . . 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)))
5251, 49eleqtrrd 2829 . . . . . . . . . 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 28872 . . . . . . . . . 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⟩)))
5441, 42, 43, 47, 50, 52, 53syl132anc 1385 . . . . . . . . 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⟩)))
5541, 11, 30, 44, 46, 48ebtwntg 28913 . . . . . . . . . 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)))
5641, 11, 30, 45, 46, 51ebtwntg 28913 . . . . . . . . . 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)))
5755, 56anbi12d 630 . . . . . . . . 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 773 . . . . . . . . . . . 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)
5948adantr 479 . . . . . . . . . . . 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)))
6045adantr 479 . . . . . . . . . . . 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 483 . . . . . . . . . . . . 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))
6249adantr 479 . . . . . . . . . . . . 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)))
6361, 62eleqtrd 2828 . . . . . . . . . . . 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)))
6458, 11, 30, 59, 60, 63ebtwntg 28913 . . . . . . . . . . 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)))
6551adantr 479 . . . . . . . . . . . 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)))
6644adantr 479 . . . . . . . . . . . 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)))
6758, 11, 30, 65, 66, 63ebtwntg 28913 . . . . . . . . . . 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)))
6864, 67anbi12d 630 . . . . . . . . . 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))))
6949, 68rexeqbidva 3318 . . . . . . . . 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))))
7054, 57, 693imtr3d 292 . . . . . . . 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))))
7170ralrimivvva 3194 . . . . . . 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))))
7271ralrimivva 3191 . . . . . 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 481 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → 𝑁 ∈ ℕ)
74 elpwi 4604 . . . . . . . . . . 11 (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) → 𝑠 ⊆ (Base‘(EEG‘𝑁)))
7574ad2antrl 726 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → 𝑠 ⊆ (Base‘(EEG‘𝑁)))
764adantr 479 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
7775, 76sseqtrrd 4020 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → 𝑠 ⊆ (𝔼‘𝑁))
78 elpwi 4604 . . . . . . . . . . 11 (𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)) → 𝑡 ⊆ (Base‘(EEG‘𝑁)))
7978ad2antll 727 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → 𝑡 ⊆ (Base‘(EEG‘𝑁)))
8079, 76sseqtrrd 4020 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → 𝑡 ⊆ (𝔼‘𝑁))
81 simpll 765 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ (𝑠 ⊆ (𝔼‘𝑁) ∧ 𝑡 ⊆ (𝔼‘𝑁))) ∧ ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥𝑠𝑦𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩) → 𝑁 ∈ ℕ)
82 simplrl 775 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ (𝑠 ⊆ (𝔼‘𝑁) ∧ 𝑡 ⊆ (𝔼‘𝑁))) ∧ ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥𝑠𝑦𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩) → 𝑠 ⊆ (𝔼‘𝑁))
83 simplrr 776 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ (𝑠 ⊆ (𝔼‘𝑁) ∧ 𝑡 ⊆ (𝔼‘𝑁))) ∧ ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥𝑠𝑦𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩) → 𝑡 ⊆ (𝔼‘𝑁))
84 simpr 483 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ (𝑠 ⊆ (𝔼‘𝑁) ∧ 𝑡 ⊆ (𝔼‘𝑁))) ∧ ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥𝑠𝑦𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩) → ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥𝑠𝑦𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩)
85 axcont 28907 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ (𝑠 ⊆ (𝔼‘𝑁) ∧ 𝑡 ⊆ (𝔼‘𝑁) ∧ ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥𝑠𝑦𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩)) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥𝑠𝑦𝑡 𝑏 Btwn ⟨𝑥, 𝑦⟩)
8681, 82, 83, 84, 85syl13anc 1369 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ (𝑠 ⊆ (𝔼‘𝑁) ∧ 𝑡 ⊆ (𝔼‘𝑁))) ∧ ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥𝑠𝑦𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥𝑠𝑦𝑡 𝑏 Btwn ⟨𝑥, 𝑦⟩)
8786ex 411 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ (𝑠 ⊆ (𝔼‘𝑁) ∧ 𝑡 ⊆ (𝔼‘𝑁))) → (∃𝑎 ∈ (𝔼‘𝑁)∀𝑥𝑠𝑦𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩ → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥𝑠𝑦𝑡 𝑏 Btwn ⟨𝑥, 𝑦⟩))
8873, 77, 80, 87syl12anc 835 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → (∃𝑎 ∈ (𝔼‘𝑁)∀𝑥𝑠𝑦𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩ → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥𝑠𝑦𝑡 𝑏 Btwn ⟨𝑥, 𝑦⟩))
89 simplll 773 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥𝑠𝑦𝑡)) → 𝑁 ∈ ℕ)
90 simplr 767 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥𝑠𝑦𝑡)) → 𝑎 ∈ (𝔼‘𝑁))
9176ad2antrr 724 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥𝑠𝑦𝑡)) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
9290, 91eleqtrd 2828 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥𝑠𝑦𝑡)) → 𝑎 ∈ (Base‘(EEG‘𝑁)))
9379ad2antrr 724 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥𝑠𝑦𝑡)) → 𝑡 ⊆ (Base‘(EEG‘𝑁)))
94 simprr 771 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥𝑠𝑦𝑡)) → 𝑦𝑡)
9593, 94sseldd 3979 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥𝑠𝑦𝑡)) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
9675ad2antrr 724 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥𝑠𝑦𝑡)) → 𝑠 ⊆ (Base‘(EEG‘𝑁)))
97 simprl 769 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥𝑠𝑦𝑡)) → 𝑥𝑠)
9896, 97sseldd 3979 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥𝑠𝑦𝑡)) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
9989, 11, 30, 92, 95, 98ebtwntg 28913 . . . . . . . . . 10 ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥𝑠𝑦𝑡)) → (𝑥 Btwn ⟨𝑎, 𝑦⟩ ↔ 𝑥 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑦)))
100992ralbidva 3207 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → (∀𝑥𝑠𝑦𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩ ↔ ∀𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑦)))
10176, 100rexeqbidva 3318 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → (∃𝑎 ∈ (𝔼‘𝑁)∀𝑥𝑠𝑦𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩ ↔ ∃𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑦)))
102 simplll 773 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥𝑠𝑦𝑡)) → 𝑁 ∈ ℕ)
10375ad2antrr 724 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥𝑠𝑦𝑡)) → 𝑠 ⊆ (Base‘(EEG‘𝑁)))
104 simprl 769 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥𝑠𝑦𝑡)) → 𝑥𝑠)
105103, 104sseldd 3979 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥𝑠𝑦𝑡)) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
10679ad2antrr 724 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥𝑠𝑦𝑡)) → 𝑡 ⊆ (Base‘(EEG‘𝑁)))
107 simprr 771 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥𝑠𝑦𝑡)) → 𝑦𝑡)
108106, 107sseldd 3979 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥𝑠𝑦𝑡)) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
109 simplr 767 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥𝑠𝑦𝑡)) → 𝑏 ∈ (𝔼‘𝑁))
11076ad2antrr 724 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥𝑠𝑦𝑡)) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
111109, 110eleqtrd 2828 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥𝑠𝑦𝑡)) → 𝑏 ∈ (Base‘(EEG‘𝑁)))
112102, 11, 30, 105, 108, 111ebtwntg 28913 . . . . . . . . . 10 ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥𝑠𝑦𝑡)) → (𝑏 Btwn ⟨𝑥, 𝑦⟩ ↔ 𝑏 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦)))
1131122ralbidva 3207 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) → (∀𝑥𝑠𝑦𝑡 𝑏 Btwn ⟨𝑥, 𝑦⟩ ↔ ∀𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦)))
11476, 113rexeqbidva 3318 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → (∃𝑏 ∈ (𝔼‘𝑁)∀𝑥𝑠𝑦𝑡 𝑏 Btwn ⟨𝑥, 𝑦⟩ ↔ ∃𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦)))
11588, 101, 1143imtr3d 292 . . . . . . 7 ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → (∃𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑦) → ∃𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦)))
116115ralrimivva 3191 . . . . . 6 (𝑁 ∈ ℕ → ∀𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁))∀𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁))(∃𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑦) → ∃𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦)))
11740, 72, 1163jca 1125 . . . . 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))∀𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑦) → ∃𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦))))
11811, 12, 30istrkgb 28379 . . . . 5 ((EEG‘𝑁) ∈ TarskiGB ↔ ((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))∀𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑦) → ∃𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦)))))
1191, 117, 118sylanbrc 581 . . . 4 (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiGB)
12032, 119elind 4192 . . 3 (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ (TarskiGC ∩ TarskiGB))
121 simplll 773 . . . . . . . . . . 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)
1223ad2antrr 724 . . . . . . . . . . . 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)))
123121, 4syl 17 . . . . . . . . . . . 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)))
124122, 123eleqtrrd 2829 . . . . . . . . . . 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))
1257ad2antrr 724 . . . . . . . . . . . 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)))
126125, 123eleqtrrd 2829 . . . . . . . . . . 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 1212 . . . . . . . . . . . 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)))
128127, 123eleqtrrd 2829 . . . . . . . . . . 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 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)))) → 𝑢 ∈ (Base‘(EEG‘𝑁)))
130129, 123eleqtrrd 2829 . . . . . . . . . . 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 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)))) → 𝑎 ∈ (Base‘(EEG‘𝑁)))
132131, 123eleqtrrd 2829 . . . . . . . . . . 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 1191 . . . . . . . . . . . 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)))
134133, 123eleqtrrd 2829 . . . . . . . . . . 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 1192 . . . . . . . . . . . 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)))
136135, 123eleqtrrd 2829 . . . . . . . . . . 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 1193 . . . . . . . . . . . 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)))
138137, 123eleqtrrd 2829 . . . . . . . . . . 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 1092 . . . . . . . . . . . 12 (((𝑥𝑦𝑦 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⟩)) ↔ ((𝑥𝑦𝑦 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 28869 . . . . . . . . . . . 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))) → (((𝑥𝑦𝑦 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⟩))
141139, 140biimtrrid 242 . . . . . . . . . . 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))) → (((𝑥𝑦𝑦 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⟩))
142121, 124, 126, 128, 130, 132, 134, 136, 138, 141syl333anc 1399 . . . . . . . . . 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)))) → (((𝑥𝑦𝑦 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⟩))
143121, 11, 30, 122, 127, 125ebtwntg 28913 . . . . . . . . . . . 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)))
144121, 11, 30, 131, 135, 133ebtwntg 28913 . . . . . . . . . . . 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)))
145143, 1443anbi23d 1436 . . . . . . . . . . 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)))) → ((𝑥𝑦𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ 𝑏 Btwn ⟨𝑎, 𝑐⟩) ↔ (𝑥𝑦𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑏 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑐))))
146121, 11, 12, 122, 125, 131, 133ecgrtg 28914 . . . . . . . . . . . . 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)))
147121, 11, 12, 125, 127, 133, 135ecgrtg 28914 . . . . . . . . . . . . 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)))
148146, 147anbi12d 630 . . . . . . . . . . . 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))))
149121, 11, 12, 122, 129, 131, 137ecgrtg 28914 . . . . . . . . . . . . 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)))
150121, 11, 12, 125, 129, 133, 137ecgrtg 28914 . . . . . . . . . . . . 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)))
151149, 150anbi12d 630 . . . . . . . . . . . 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))))
152148, 151anbi12d 630 . . . . . . . . . . 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)))))
153145, 152anbi12d 630 . . . . . . . . . 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)))) → (((𝑥𝑦𝑦 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⟩))) ↔ ((𝑥𝑦𝑦 ∈ (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))))))
154121, 11, 12, 127, 129, 135, 137ecgrtg 28914 . . . . . . . . . 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)))
155142, 153, 1543imtr3d 292 . . . . . . . . 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)))) → (((𝑥𝑦𝑦 ∈ (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)))
156155ralrimivvva 3194 . . . . . . . 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))(((𝑥𝑦𝑦 ∈ (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)))
157156ralrimivvva 3194 . . . . . . 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))(((𝑥𝑦𝑦 ∈ (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)))
158157ralrimivva 3191 . . . . . 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))(((𝑥𝑦𝑦 ∈ (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 765 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → 𝑁 ∈ ℕ)
1606adantr 479 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (𝔼‘𝑁))
1618adantr 479 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (𝔼‘𝑁))
162 simprl 769 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → 𝑎 ∈ (Base‘(EEG‘𝑁)))
163159, 4syl 17 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
164162, 163eleqtrrd 2829 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → 𝑎 ∈ (𝔼‘𝑁))
165 simprr 771 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → 𝑏 ∈ (Base‘(EEG‘𝑁)))
166165, 163eleqtrrd 2829 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → 𝑏 ∈ (𝔼‘𝑁))
167 axsegcon 28858 . . . . . . . . . 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⟩))
168159, 160, 161, 164, 166, 167syl122anc 1376 . . . . . . . . 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 773 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
1703ad2antrr 724 . . . . . . . . . . . 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 483 . . . . . . . . . . . . 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))
172163adantr 479 . . . . . . . . . . . . 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)))
173171, 172eleqtrd 2828 . . . . . . . . . . . 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)))
1747ad2antrr 724 . . . . . . . . . . . 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)))
175169, 11, 30, 170, 173, 174ebtwntg 28913 . . . . . . . . . . 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 775 . . . . . . . . . . . 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 776 . . . . . . . . . . . 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)))
178169, 11, 12, 174, 173, 176, 177ecgrtg 28914 . . . . . . . . . . 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)))
179175, 178anbi12d 630 . . . . . . . . . 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))))
180163, 179rexeqbidva 3318 . . . . . . . . 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))))
181168, 180mpbid 231 . . . . . . . 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)))
182181ralrimivva 3191 . . . . . . 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)))
183182ralrimivva 3191 . . . . . 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)))
1841, 158, 183jca32 514 . . . . 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))(((𝑥𝑦𝑦 ∈ (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)))))
18511, 12, 30istrkgcb 28380 . . . . 5 ((EEG‘𝑁) ∈ TarskiGCB ↔ ((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))(((𝑥𝑦𝑦 ∈ (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)))))
186184, 185sylibr 233 . . . 4 (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiGCB)
18711, 30elntg 28915 . . . . 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))}))
18811, 12, 30istrkgl 28382 . . . . 5 ((EEG‘𝑁) ∈ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (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))})))
1891, 187, 188sylanbrc 581 . . . 4 (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})
190186, 189elind 4192 . . 3 (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
191120, 190elind 4192 . 2 (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})))
192 df-trkg 28377 . 2 TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
193191, 192eleqtrrdi 2837 1 (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3o 1083  w3a 1084   = wceq 1534  wcel 2099  {cab 2703  wne 2930  wral 3051  wrex 3060  {crab 3419  Vcvv 3462  [wsbc 3775  cdif 3943  cin 3945  wss 3946  𝒫 cpw 4597  {csn 4623  cop 4629   class class class wbr 5145  cfv 6546  (class class class)co 7416  cmpo 7418  cn 12258  Basecbs 17208  distcds 17270  TarskiGcstrkg 28351  TarskiGCcstrkgc 28352  TarskiGBcstrkgb 28353  TarskiGCBcstrkgcb 28354  Itvcitv 28357  LineGclng 28358  𝔼cee 28819   Btwn cbtwn 28820  Cgrccgr 28821  EEGceeng 28908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-inf2 9677  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226  ax-pre-sup 11227
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-se 5630  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-isom 6555  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7995  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-er 8726  df-map 8849  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-sup 9478  df-oi 9546  df-card 9975  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11487  df-neg 11488  df-div 11913  df-nn 12259  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-7 12326  df-8 12327  df-9 12328  df-n0 12519  df-z 12605  df-dec 12724  df-uz 12869  df-rp 13023  df-ico 13378  df-icc 13379  df-fz 13533  df-fzo 13676  df-seq 14016  df-exp 14076  df-hash 14343  df-cj 15099  df-re 15100  df-im 15101  df-sqrt 15235  df-abs 15236  df-clim 15485  df-sum 15686  df-struct 17144  df-slot 17179  df-ndx 17191  df-base 17209  df-ds 17283  df-itv 28359  df-lng 28360  df-trkgc 28372  df-trkgb 28373  df-trkgcb 28374  df-trkg 28377  df-ee 28822  df-btwn 28823  df-cgr 28824  df-eeng 28909
This theorem is referenced by: (None)
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