Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  istrkg2d Structured version   Visualization version   GIF version

Theorem istrkg2d 33279
Description: Property of fulfilling dimension 2 axiom. (Contributed by Thierry Arnoux, 29-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
istrkg2d.p 𝑃 = (Base‘𝐺)
istrkg2d.d = (dist‘𝐺)
istrkg2d.i 𝐼 = (Itv‘𝐺)
Assertion
Ref Expression
istrkg2d (𝐺 ∈ TarskiG2D ↔ (𝐺 ∈ V ∧ (∃𝑥𝑃𝑦𝑃𝑧𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
Distinct variable groups:   𝑢, ,𝑣,𝑥,𝑦,𝑧   𝑢,𝐼,𝑣,𝑥,𝑦,𝑧   𝑢,𝑃,𝑣,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐺(𝑥,𝑦,𝑧,𝑣,𝑢)

Proof of Theorem istrkg2d
Dummy variables 𝑑 𝑓 𝑖 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 istrkg2d.p . . 3 𝑃 = (Base‘𝐺)
2 istrkg2d.d . . 3 = (dist‘𝐺)
3 istrkg2d.i . . 3 𝐼 = (Itv‘𝐺)
4 simp1 1136 . . . . . 6 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑝 = 𝑃)
54eqcomd 2742 . . . . 5 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑃 = 𝑝)
6 simp3 1138 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑖 = 𝐼)
76eqcomd 2742 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝐼 = 𝑖)
87oveqd 7374 . . . . . . . . . 10 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑥𝐼𝑦) = (𝑥𝑖𝑦))
98eleq2d 2823 . . . . . . . . 9 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑥𝑖𝑦)))
107oveqd 7374 . . . . . . . . . 10 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑧𝐼𝑦) = (𝑧𝑖𝑦))
1110eleq2d 2823 . . . . . . . . 9 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝑥 ∈ (𝑧𝑖𝑦)))
127oveqd 7374 . . . . . . . . . 10 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑥𝐼𝑧) = (𝑥𝑖𝑧))
1312eleq2d 2823 . . . . . . . . 9 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑥𝑖𝑧)))
149, 11, 133orbi123d 1435 . . . . . . . 8 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))
1514notbid 317 . . . . . . 7 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))
165, 15rexeqbidv 3320 . . . . . 6 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (∃𝑧𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ ∃𝑧𝑝 ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))
175, 16rexeqbidv 3320 . . . . 5 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (∃𝑦𝑃𝑧𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ ∃𝑦𝑝𝑧𝑝 ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))
185, 17rexeqbidv 3320 . . . 4 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (∃𝑥𝑃𝑦𝑃𝑧𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ ∃𝑥𝑝𝑦𝑝𝑧𝑝 ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))
19 simp2 1137 . . . . . . . . . . . . . . 15 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑑 = )
2019eqcomd 2742 . . . . . . . . . . . . . 14 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → = 𝑑)
2120oveqd 7374 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑥 𝑢) = (𝑥𝑑𝑢))
2220oveqd 7374 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑥 𝑣) = (𝑥𝑑𝑣))
2321, 22eqeq12d 2752 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑥 𝑢) = (𝑥 𝑣) ↔ (𝑥𝑑𝑢) = (𝑥𝑑𝑣)))
2420oveqd 7374 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑦 𝑢) = (𝑦𝑑𝑢))
2520oveqd 7374 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑦 𝑣) = (𝑦𝑑𝑣))
2624, 25eqeq12d 2752 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑦 𝑢) = (𝑦 𝑣) ↔ (𝑦𝑑𝑢) = (𝑦𝑑𝑣)))
2720oveqd 7374 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑧 𝑢) = (𝑧𝑑𝑢))
2820oveqd 7374 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑧 𝑣) = (𝑧𝑑𝑣))
2927, 28eqeq12d 2752 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑧 𝑢) = (𝑧 𝑣) ↔ (𝑧𝑑𝑢) = (𝑧𝑑𝑣)))
3023, 26, 293anbi123d 1436 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ↔ ((𝑥𝑑𝑢) = (𝑥𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑦𝑑𝑣) ∧ (𝑧𝑑𝑢) = (𝑧𝑑𝑣))))
3130anbi1d 630 . . . . . . . . . 10 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) ↔ (((𝑥𝑑𝑢) = (𝑥𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑦𝑑𝑣) ∧ (𝑧𝑑𝑢) = (𝑧𝑑𝑣)) ∧ 𝑢𝑣)))
3231, 14imbi12d 344 . . . . . . . . 9 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (((((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ((((𝑥𝑑𝑢) = (𝑥𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑦𝑑𝑣) ∧ (𝑧𝑑𝑢) = (𝑧𝑑𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)))))
335, 32raleqbidv 3319 . . . . . . . 8 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (∀𝑣𝑃 ((((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ∀𝑣𝑝 ((((𝑥𝑑𝑢) = (𝑥𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑦𝑑𝑣) ∧ (𝑧𝑑𝑢) = (𝑧𝑑𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)))))
345, 33raleqbidv 3319 . . . . . . 7 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (∀𝑢𝑃𝑣𝑃 ((((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ∀𝑢𝑝𝑣𝑝 ((((𝑥𝑑𝑢) = (𝑥𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑦𝑑𝑣) ∧ (𝑧𝑑𝑢) = (𝑧𝑑𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)))))
355, 34raleqbidv 3319 . . . . . 6 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (∀𝑧𝑃𝑢𝑃𝑣𝑃 ((((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ∀𝑧𝑝𝑢𝑝𝑣𝑝 ((((𝑥𝑑𝑢) = (𝑥𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑦𝑑𝑣) ∧ (𝑧𝑑𝑢) = (𝑧𝑑𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)))))
365, 35raleqbidv 3319 . . . . 5 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (∀𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ∀𝑦𝑝𝑧𝑝𝑢𝑝𝑣𝑝 ((((𝑥𝑑𝑢) = (𝑥𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑦𝑑𝑣) ∧ (𝑧𝑑𝑢) = (𝑧𝑑𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)))))
375, 36raleqbidv 3319 . . . 4 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ∀𝑥𝑝𝑦𝑝𝑧𝑝𝑢𝑝𝑣𝑝 ((((𝑥𝑑𝑢) = (𝑥𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑦𝑑𝑣) ∧ (𝑧𝑑𝑢) = (𝑧𝑑𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)))))
3818, 37anbi12d 631 . . 3 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((∃𝑥𝑃𝑦𝑃𝑧𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))) ↔ (∃𝑥𝑝𝑦𝑝𝑧𝑝 ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝𝑢𝑝𝑣𝑝 ((((𝑥𝑑𝑢) = (𝑥𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑦𝑑𝑣) ∧ (𝑧𝑑𝑢) = (𝑧𝑑𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))))
391, 2, 3, 38sbcie3s 17034 . 2 (𝑓 = 𝐺 → ([(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑][(Itv‘𝑓) / 𝑖](∃𝑥𝑝𝑦𝑝𝑧𝑝 ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝𝑢𝑝𝑣𝑝 ((((𝑥𝑑𝑢) = (𝑥𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑦𝑑𝑣) ∧ (𝑧𝑑𝑢) = (𝑧𝑑𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)))) ↔ (∃𝑥𝑃𝑦𝑃𝑧𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
40 df-trkg2d 33278 . 2 TarskiG2D = {𝑓[(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑][(Itv‘𝑓) / 𝑖](∃𝑥𝑝𝑦𝑝𝑧𝑝 ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝𝑢𝑝𝑣𝑝 ((((𝑥𝑑𝑢) = (𝑥𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑦𝑑𝑣) ∧ (𝑧𝑑𝑢) = (𝑧𝑑𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))}
4139, 40elab4g 3635 1 (𝐺 ∈ TarskiG2D ↔ (𝐺 ∈ V ∧ (∃𝑥𝑃𝑦𝑃𝑧𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3o 1086  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  Vcvv 3445  [wsbc 3739  cfv 6496  (class class class)co 7357  Basecbs 17083  distcds 17142  Itvcitv 27375  TarskiG2Dcstrkg2d 33277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-nul 5263
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-iota 6448  df-fv 6504  df-ov 7360  df-trkg2d 33278
This theorem is referenced by:  axtglowdim2ALTV  33280  axtgupdim2ALTV  33281
  Copyright terms: Public domain W3C validator