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Theorem ntrk2imkb 43532
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 6892 . . . . . 6 (𝑠 = 𝑡 → (𝐼𝑠) = (𝐼𝑡))
3 id 22 . . . . . 6 (𝑠 = 𝑡𝑠 = 𝑡)
42, 3sseq12d 4006 . . . . 5 (𝑠 = 𝑡 → ((𝐼𝑠) ⊆ 𝑠 ↔ (𝐼𝑡) ⊆ 𝑡))
54cbvralvw 3225 . . . 4 (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐼𝑡) ⊆ 𝑡)
65biimpi 215 . . 3 (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 → ∀𝑡 ∈ 𝒫 𝐵(𝐼𝑡) ⊆ 𝑡)
7 raaanv 4517 . . 3 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) ↔ (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 ∧ ∀𝑡 ∈ 𝒫 𝐵(𝐼𝑡) ⊆ 𝑡))
81, 6, 7sylanbrc 581 . 2 (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 → ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡))
9 ss2in 4231 . . . . . . 7 (((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) → ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝑠𝑡))
109adantr 479 . . . . . 6 ((((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) ∧ (𝑠𝑡) = ∅) → ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝑠𝑡))
11 simpr 483 . . . . . 6 ((((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) ∧ (𝑠𝑡) = ∅) → (𝑠𝑡) = ∅)
1210, 11sseqtrd 4013 . . . . 5 ((((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) ∧ (𝑠𝑡) = ∅) → ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ ∅)
13 ss0 4394 . . . . 5 (((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)
1412, 13syl 17 . . . 4 ((((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) ∧ (𝑠𝑡) = ∅) → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)
1514ex 411 . . 3 (((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) → ((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅))
16152ralimi 3113 . 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 394   = wceq 1533  wral 3051  cin 3938  wss 3939  c0 4318  𝒫 cpw 4598  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-iota 6495  df-fv 6551
This theorem is referenced by: (None)
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