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Theorem exmidsbthrlem 13424
 Description: Lemma for exmidsbthr 13425. (Contributed by Jim Kingdon, 11-Aug-2022.)
Hypothesis
Ref Expression
exmidsbthrlem.s 𝑆 = (𝑝 ∈ ℕ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))))
Assertion
Ref Expression
exmidsbthrlem (∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) → EXMID)
Distinct variable groups:   𝑆,𝑖   𝑖,𝑝   𝑥,𝑦
Allowed substitution hints:   𝑆(𝑥,𝑦,𝑝)

Proof of Theorem exmidsbthrlem
Dummy variables 𝑎 𝑏 𝑘 𝑧 𝑓 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 108 . . . . . . 7 ((∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) ∧ 𝑧 ⊆ {∅}) → ∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦))
2 nninfex 13412 . . . . . . . . . 10 ∈ V
3 fconstmpt 4596 . . . . . . . . . . . . . . 15 (ω × {∅}) = (𝑖 ∈ ω ↦ ∅)
4 0nninf 13404 . . . . . . . . . . . . . . 15 (ω × {∅}) ∈ ℕ
53, 4eqeltrri 2214 . . . . . . . . . . . . . 14 (𝑖 ∈ ω ↦ ∅) ∈ ℕ
65fconst6 5332 . . . . . . . . . . . . 13 (𝑧 × {(𝑖 ∈ ω ↦ ∅)}):𝑧⟶ℕ
76a1i 9 . . . . . . . . . . . 12 (𝑧 ⊆ {∅} → (𝑧 × {(𝑖 ∈ ω ↦ ∅)}):𝑧⟶ℕ)
8 ssel 3097 . . . . . . . . . . . . . . . . . 18 (𝑧 ⊆ {∅} → (𝑢𝑧𝑢 ∈ {∅}))
9 elsni 3551 . . . . . . . . . . . . . . . . . 18 (𝑢 ∈ {∅} → 𝑢 = ∅)
108, 9syl6 33 . . . . . . . . . . . . . . . . 17 (𝑧 ⊆ {∅} → (𝑢𝑧𝑢 = ∅))
11 ssel 3097 . . . . . . . . . . . . . . . . . 18 (𝑧 ⊆ {∅} → (𝑣𝑧𝑣 ∈ {∅}))
12 elsni 3551 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ {∅} → 𝑣 = ∅)
1311, 12syl6 33 . . . . . . . . . . . . . . . . 17 (𝑧 ⊆ {∅} → (𝑣𝑧𝑣 = ∅))
1410, 13anim12d 333 . . . . . . . . . . . . . . . 16 (𝑧 ⊆ {∅} → ((𝑢𝑧𝑣𝑧) → (𝑢 = ∅ ∧ 𝑣 = ∅)))
15 eqtr3 2160 . . . . . . . . . . . . . . . 16 ((𝑢 = ∅ ∧ 𝑣 = ∅) → 𝑢 = 𝑣)
1614, 15syl6 33 . . . . . . . . . . . . . . 15 (𝑧 ⊆ {∅} → ((𝑢𝑧𝑣𝑧) → 𝑢 = 𝑣))
1716imp 123 . . . . . . . . . . . . . 14 ((𝑧 ⊆ {∅} ∧ (𝑢𝑧𝑣𝑧)) → 𝑢 = 𝑣)
1817a1d 22 . . . . . . . . . . . . 13 ((𝑧 ⊆ {∅} ∧ (𝑢𝑧𝑣𝑧)) → (((𝑧 × {(𝑖 ∈ ω ↦ ∅)})‘𝑢) = ((𝑧 × {(𝑖 ∈ ω ↦ ∅)})‘𝑣) → 𝑢 = 𝑣))
1918ralrimivva 2518 . . . . . . . . . . . 12 (𝑧 ⊆ {∅} → ∀𝑢𝑧𝑣𝑧 (((𝑧 × {(𝑖 ∈ ω ↦ ∅)})‘𝑢) = ((𝑧 × {(𝑖 ∈ ω ↦ ∅)})‘𝑣) → 𝑢 = 𝑣))
20 dff13 5679 . . . . . . . . . . . 12 ((𝑧 × {(𝑖 ∈ ω ↦ ∅)}):𝑧1-1→ℕ ↔ ((𝑧 × {(𝑖 ∈ ω ↦ ∅)}):𝑧⟶ℕ ∧ ∀𝑢𝑧𝑣𝑧 (((𝑧 × {(𝑖 ∈ ω ↦ ∅)})‘𝑢) = ((𝑧 × {(𝑖 ∈ ω ↦ ∅)})‘𝑣) → 𝑢 = 𝑣)))
217, 19, 20sylanbrc 414 . . . . . . . . . . 11 (𝑧 ⊆ {∅} → (𝑧 × {(𝑖 ∈ ω ↦ ∅)}):𝑧1-1→ℕ)
22 exmidsbthrlem.s . . . . . . . . . . . . 13 𝑆 = (𝑝 ∈ ℕ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))))
2322peano4nninf 13407 . . . . . . . . . . . 12 𝑆:ℕ1-1→ℕ
2423a1i 9 . . . . . . . . . . 11 (𝑧 ⊆ {∅} → 𝑆:ℕ1-1→ℕ)
25 disj 3417 . . . . . . . . . . . . 13 ((ran (𝑧 × {(𝑖 ∈ ω ↦ ∅)}) ∩ ran 𝑆) = ∅ ↔ ∀𝑎 ∈ ran (𝑧 × {(𝑖 ∈ ω ↦ ∅)}) ¬ 𝑎 ∈ ran 𝑆)
2622peano3nninf 13408 . . . . . . . . . . . . . . . . . . 19 (𝑏 ∈ ℕ → (𝑆𝑏) ≠ (𝑘 ∈ ω ↦ ∅))
27 eqidd 2141 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑖 → ∅ = ∅)
2827cbvmptv 4033 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ω ↦ ∅) = (𝑖 ∈ ω ↦ ∅)
2928neeq2i 2325 . . . . . . . . . . . . . . . . . . 19 ((𝑆𝑏) ≠ (𝑘 ∈ ω ↦ ∅) ↔ (𝑆𝑏) ≠ (𝑖 ∈ ω ↦ ∅))
3026, 29sylib 121 . . . . . . . . . . . . . . . . . 18 (𝑏 ∈ ℕ → (𝑆𝑏) ≠ (𝑖 ∈ ω ↦ ∅))
3130neneqd 2330 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ ℕ → ¬ (𝑆𝑏) = (𝑖 ∈ ω ↦ ∅))
3231nrex 2528 . . . . . . . . . . . . . . . 16 ¬ ∃𝑏 ∈ ℕ (𝑆𝑏) = (𝑖 ∈ ω ↦ ∅)
33 f1dm 5343 . . . . . . . . . . . . . . . . . 18 (𝑆:ℕ1-1→ℕ → dom 𝑆 = ℕ)
3423, 33ax-mp 5 . . . . . . . . . . . . . . . . 17 dom 𝑆 = ℕ
35 eqcom 2142 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ ω ↦ ∅) = (𝑆𝑏) ↔ (𝑆𝑏) = (𝑖 ∈ ω ↦ ∅))
3634, 35rexeqbii 2452 . . . . . . . . . . . . . . . 16 (∃𝑏 ∈ dom 𝑆(𝑖 ∈ ω ↦ ∅) = (𝑆𝑏) ↔ ∃𝑏 ∈ ℕ (𝑆𝑏) = (𝑖 ∈ ω ↦ ∅))
3732, 36mtbir 661 . . . . . . . . . . . . . . 15 ¬ ∃𝑏 ∈ dom 𝑆(𝑖 ∈ ω ↦ ∅) = (𝑆𝑏)
3822funmpt2 5172 . . . . . . . . . . . . . . . 16 Fun 𝑆
39 elrnrexdm 5569 . . . . . . . . . . . . . . . 16 (Fun 𝑆 → ((𝑖 ∈ ω ↦ ∅) ∈ ran 𝑆 → ∃𝑏 ∈ dom 𝑆(𝑖 ∈ ω ↦ ∅) = (𝑆𝑏)))
4038, 39ax-mp 5 . . . . . . . . . . . . . . 15 ((𝑖 ∈ ω ↦ ∅) ∈ ran 𝑆 → ∃𝑏 ∈ dom 𝑆(𝑖 ∈ ω ↦ ∅) = (𝑆𝑏))
4137, 40mto 652 . . . . . . . . . . . . . 14 ¬ (𝑖 ∈ ω ↦ ∅) ∈ ran 𝑆
42 rnxpss 4980 . . . . . . . . . . . . . . . . 17 ran (𝑧 × {(𝑖 ∈ ω ↦ ∅)}) ⊆ {(𝑖 ∈ ω ↦ ∅)}
4342sseli 3099 . . . . . . . . . . . . . . . 16 (𝑎 ∈ ran (𝑧 × {(𝑖 ∈ ω ↦ ∅)}) → 𝑎 ∈ {(𝑖 ∈ ω ↦ ∅)})
44 elsni 3551 . . . . . . . . . . . . . . . 16 (𝑎 ∈ {(𝑖 ∈ ω ↦ ∅)} → 𝑎 = (𝑖 ∈ ω ↦ ∅))
4543, 44syl 14 . . . . . . . . . . . . . . 15 (𝑎 ∈ ran (𝑧 × {(𝑖 ∈ ω ↦ ∅)}) → 𝑎 = (𝑖 ∈ ω ↦ ∅))
4645eleq1d 2209 . . . . . . . . . . . . . 14 (𝑎 ∈ ran (𝑧 × {(𝑖 ∈ ω ↦ ∅)}) → (𝑎 ∈ ran 𝑆 ↔ (𝑖 ∈ ω ↦ ∅) ∈ ran 𝑆))
4741, 46mtbiri 665 . . . . . . . . . . . . 13 (𝑎 ∈ ran (𝑧 × {(𝑖 ∈ ω ↦ ∅)}) → ¬ 𝑎 ∈ ran 𝑆)
4825, 47mprgbir 2494 . . . . . . . . . . . 12 (ran (𝑧 × {(𝑖 ∈ ω ↦ ∅)}) ∩ ran 𝑆) = ∅
4948a1i 9 . . . . . . . . . . 11 (𝑧 ⊆ {∅} → (ran (𝑧 × {(𝑖 ∈ ω ↦ ∅)}) ∩ ran 𝑆) = ∅)
5021, 24, 49casef1 6988 . . . . . . . . . 10 (𝑧 ⊆ {∅} → case((𝑧 × {(𝑖 ∈ ω ↦ ∅)}), 𝑆):(𝑧 ⊔ ℕ)–1-1→ℕ)
51 f1domg 6662 . . . . . . . . . 10 (ℕ ∈ V → (case((𝑧 × {(𝑖 ∈ ω ↦ ∅)}), 𝑆):(𝑧 ⊔ ℕ)–1-1→ℕ → (𝑧 ⊔ ℕ) ≼ ℕ))
522, 50, 51mpsyl 65 . . . . . . . . 9 (𝑧 ⊆ {∅} → (𝑧 ⊔ ℕ) ≼ ℕ)
5352adantl 275 . . . . . . . 8 ((∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) ∧ 𝑧 ⊆ {∅}) → (𝑧 ⊔ ℕ) ≼ ℕ)
54 inrresf1 6960 . . . . . . . . 9 (inr ↾ ℕ):ℕ1-1→(𝑧 ⊔ ℕ)
55 vex 2693 . . . . . . . . . . 11 𝑧 ∈ V
56 djuex 6941 . . . . . . . . . . 11 ((𝑧 ∈ V ∧ ℕ ∈ V) → (𝑧 ⊔ ℕ) ∈ V)
5755, 2, 56mp2an 423 . . . . . . . . . 10 (𝑧 ⊔ ℕ) ∈ V
5857f1dom 6664 . . . . . . . . 9 ((inr ↾ ℕ):ℕ1-1→(𝑧 ⊔ ℕ) → ℕ ≼ (𝑧 ⊔ ℕ))
5954, 58ax-mp 5 . . . . . . . 8 ≼ (𝑧 ⊔ ℕ)
6053, 59jctir 311 . . . . . . 7 ((∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) ∧ 𝑧 ⊆ {∅}) → ((𝑧 ⊔ ℕ) ≼ ℕ ∧ ℕ ≼ (𝑧 ⊔ ℕ)))
61 breq12 3943 . . . . . . . . . . 11 ((𝑥 = (𝑧 ⊔ ℕ) ∧ 𝑦 = ℕ) → (𝑥𝑦 ↔ (𝑧 ⊔ ℕ) ≼ ℕ))
62 breq12 3943 . . . . . . . . . . . 12 ((𝑦 = ℕ𝑥 = (𝑧 ⊔ ℕ)) → (𝑦𝑥 ↔ ℕ ≼ (𝑧 ⊔ ℕ)))
6362ancoms 266 . . . . . . . . . . 11 ((𝑥 = (𝑧 ⊔ ℕ) ∧ 𝑦 = ℕ) → (𝑦𝑥 ↔ ℕ ≼ (𝑧 ⊔ ℕ)))
6461, 63anbi12d 465 . . . . . . . . . 10 ((𝑥 = (𝑧 ⊔ ℕ) ∧ 𝑦 = ℕ) → ((𝑥𝑦𝑦𝑥) ↔ ((𝑧 ⊔ ℕ) ≼ ℕ ∧ ℕ ≼ (𝑧 ⊔ ℕ))))
65 breq12 3943 . . . . . . . . . 10 ((𝑥 = (𝑧 ⊔ ℕ) ∧ 𝑦 = ℕ) → (𝑥𝑦 ↔ (𝑧 ⊔ ℕ) ≈ ℕ))
6664, 65imbi12d 233 . . . . . . . . 9 ((𝑥 = (𝑧 ⊔ ℕ) ∧ 𝑦 = ℕ) → (((𝑥𝑦𝑦𝑥) → 𝑥𝑦) ↔ (((𝑧 ⊔ ℕ) ≼ ℕ ∧ ℕ ≼ (𝑧 ⊔ ℕ)) → (𝑧 ⊔ ℕ) ≈ ℕ)))
6766spc2gv 2781 . . . . . . . 8 (((𝑧 ⊔ ℕ) ∈ V ∧ ℕ ∈ V) → (∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) → (((𝑧 ⊔ ℕ) ≼ ℕ ∧ ℕ ≼ (𝑧 ⊔ ℕ)) → (𝑧 ⊔ ℕ) ≈ ℕ)))
6857, 2, 67mp2an 423 . . . . . . 7 (∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) → (((𝑧 ⊔ ℕ) ≼ ℕ ∧ ℕ ≼ (𝑧 ⊔ ℕ)) → (𝑧 ⊔ ℕ) ≈ ℕ))
691, 60, 68sylc 62 . . . . . 6 ((∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) ∧ 𝑧 ⊆ {∅}) → (𝑧 ⊔ ℕ) ≈ ℕ)
70 bren 6651 . . . . . 6 ((𝑧 ⊔ ℕ) ≈ ℕ ↔ ∃𝑓 𝑓:(𝑧 ⊔ ℕ)–1-1-onto→ℕ)
7169, 70sylib 121 . . . . 5 ((∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) ∧ 𝑧 ⊆ {∅}) → ∃𝑓 𝑓:(𝑧 ⊔ ℕ)–1-1-onto→ℕ)
72 nninfomni 13422 . . . . . . . . 9 ∈ Omni
7372a1i 9 . . . . . . . 8 (((∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) ∧ 𝑧 ⊆ {∅}) ∧ 𝑓:(𝑧 ⊔ ℕ)–1-1-onto→ℕ) → ℕ ∈ Omni)
74 f1ocnv 5390 . . . . . . . . . 10 (𝑓:(𝑧 ⊔ ℕ)–1-1-onto→ℕ𝑓:ℕ1-1-onto→(𝑧 ⊔ ℕ))
75 f1ofo 5384 . . . . . . . . . 10 (𝑓:ℕ1-1-onto→(𝑧 ⊔ ℕ) → 𝑓:ℕonto→(𝑧 ⊔ ℕ))
7674, 75syl 14 . . . . . . . . 9 (𝑓:(𝑧 ⊔ ℕ)–1-1-onto→ℕ𝑓:ℕonto→(𝑧 ⊔ ℕ))
7776adantl 275 . . . . . . . 8 (((∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) ∧ 𝑧 ⊆ {∅}) ∧ 𝑓:(𝑧 ⊔ ℕ)–1-1-onto→ℕ) → 𝑓:ℕonto→(𝑧 ⊔ ℕ))
7873, 77fodjuomni 7034 . . . . . . 7 (((∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) ∧ 𝑧 ⊆ {∅}) ∧ 𝑓:(𝑧 ⊔ ℕ)–1-1-onto→ℕ) → (∃𝑤 𝑤𝑧𝑧 = ∅))
79 sssnm 3690 . . . . . . . . . 10 (∃𝑤 𝑤𝑧 → (𝑧 ⊆ {∅} ↔ 𝑧 = {∅}))
8079biimpcd 158 . . . . . . . . 9 (𝑧 ⊆ {∅} → (∃𝑤 𝑤𝑧𝑧 = {∅}))
8180ad2antlr 481 . . . . . . . 8 (((∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) ∧ 𝑧 ⊆ {∅}) ∧ 𝑓:(𝑧 ⊔ ℕ)–1-1-onto→ℕ) → (∃𝑤 𝑤𝑧𝑧 = {∅}))
8281orim1d 777 . . . . . . 7 (((∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) ∧ 𝑧 ⊆ {∅}) ∧ 𝑓:(𝑧 ⊔ ℕ)–1-1-onto→ℕ) → ((∃𝑤 𝑤𝑧𝑧 = ∅) → (𝑧 = {∅} ∨ 𝑧 = ∅)))
8378, 82mpd 13 . . . . . 6 (((∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) ∧ 𝑧 ⊆ {∅}) ∧ 𝑓:(𝑧 ⊔ ℕ)–1-1-onto→ℕ) → (𝑧 = {∅} ∨ 𝑧 = ∅))
8483orcomd 719 . . . . 5 (((∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) ∧ 𝑧 ⊆ {∅}) ∧ 𝑓:(𝑧 ⊔ ℕ)–1-1-onto→ℕ) → (𝑧 = ∅ ∨ 𝑧 = {∅}))
8571, 84exlimddv 1871 . . . 4 ((∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) ∧ 𝑧 ⊆ {∅}) → (𝑧 = ∅ ∨ 𝑧 = {∅}))
8685ex 114 . . 3 (∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) → (𝑧 ⊆ {∅} → (𝑧 = ∅ ∨ 𝑧 = {∅})))
8786alrimiv 1847 . 2 (∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) → ∀𝑧(𝑧 ⊆ {∅} → (𝑧 = ∅ ∨ 𝑧 = {∅})))
88 exmid01 4130 . 2 (EXMID ↔ ∀𝑧(𝑧 ⊆ {∅} → (𝑧 = ∅ ∨ 𝑧 = {∅})))
8987, 88sylibr 133 1 (∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) → EXMID)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698  ∀wal 1330   = wceq 1332  ∃wex 1469   ∈ wcel 1481   ≠ wne 2309  ∀wral 2417  ∃wrex 2418  Vcvv 2690   ∩ cin 3076   ⊆ wss 3077  ∅c0 3369  ifcif 3480  {csn 3533  ∪ cuni 3745   class class class wbr 3938   ↦ cmpt 3998  EXMIDwem 4127  ωcom 4513   × cxp 4547  ◡ccnv 4548  dom cdm 4549  ran crn 4550   ↾ cres 4551  Fun wfun 5127  ⟶wf 5129  –1-1→wf1 5130  –onto→wfo 5131  –1-1-onto→wf1o 5132  ‘cfv 5133  1oc1o 6316   ≈ cen 6642   ≼ cdom 6643   ⊔ cdju 6935  inrcinr 6944  casecdjucase 6981  Omnicomni 7017  ℕ∞xnninf 7018 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4052  ax-sep 4055  ax-nul 4063  ax-pow 4107  ax-pr 4140  ax-un 4364  ax-setind 4461  ax-iinf 4511 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2692  df-sbc 2915  df-csb 3009  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-nul 3370  df-if 3481  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-int 3781  df-iun 3824  df-br 3939  df-opab 3999  df-mpt 4000  df-tr 4036  df-exmid 4128  df-id 4224  df-iord 4297  df-on 4299  df-suc 4302  df-iom 4514  df-xp 4555  df-rel 4556  df-cnv 4557  df-co 4558  df-dm 4559  df-rn 4560  df-res 4561  df-ima 4562  df-iota 5098  df-fun 5135  df-fn 5136  df-f 5137  df-f1 5138  df-fo 5139  df-f1o 5140  df-fv 5141  df-ov 5787  df-oprab 5788  df-mpo 5789  df-1st 6048  df-2nd 6049  df-1o 6323  df-2o 6324  df-map 6554  df-en 6645  df-dom 6646  df-dju 6936  df-inl 6945  df-inr 6946  df-case 6982  df-omni 7019  df-nninf 7020 This theorem is referenced by:  exmidsbthr  13425
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