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Theorem ntrk2imkb 44050
Description: If an interior function is contracting, the interiors of disjoint sets are disjoint. Kuratowski's K2 axiom implies KB. Interior version. (Contributed by RP, 9-Jun-2021.)
Assertion
Ref Expression
ntrk2imkb (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 → ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅))
Distinct variable groups:   𝐵,𝑠,𝑡   𝐼,𝑠,𝑡

Proof of Theorem ntrk2imkb
StepHypRef Expression
1 id 22 . . 3 (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 → ∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠)
2 fveq2 6906 . . . . . 6 (𝑠 = 𝑡 → (𝐼𝑠) = (𝐼𝑡))
3 id 22 . . . . . 6 (𝑠 = 𝑡𝑠 = 𝑡)
42, 3sseq12d 4017 . . . . 5 (𝑠 = 𝑡 → ((𝐼𝑠) ⊆ 𝑠 ↔ (𝐼𝑡) ⊆ 𝑡))
54cbvralvw 3237 . . . 4 (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐼𝑡) ⊆ 𝑡)
65biimpi 216 . . 3 (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 → ∀𝑡 ∈ 𝒫 𝐵(𝐼𝑡) ⊆ 𝑡)
7 raaanv 4518 . . 3 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) ↔ (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 ∧ ∀𝑡 ∈ 𝒫 𝐵(𝐼𝑡) ⊆ 𝑡))
81, 6, 7sylanbrc 583 . 2 (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 → ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡))
9 ss2in 4245 . . . . . . 7 (((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) → ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝑠𝑡))
109adantr 480 . . . . . 6 ((((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) ∧ (𝑠𝑡) = ∅) → ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝑠𝑡))
11 simpr 484 . . . . . 6 ((((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) ∧ (𝑠𝑡) = ∅) → (𝑠𝑡) = ∅)
1210, 11sseqtrd 4020 . . . . 5 ((((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) ∧ (𝑠𝑡) = ∅) → ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ ∅)
13 ss0 4402 . . . . 5 (((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)
1412, 13syl 17 . . . 4 ((((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) ∧ (𝑠𝑡) = ∅) → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)
1514ex 412 . . 3 (((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) → ((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅))
16152ralimi 3123 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) → ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅))
178, 16syl 17 1 (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 → ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wral 3061  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569
This theorem is referenced by: (None)
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