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Theorem ntrk2imkb 40265
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 6663 . . . . . 6 (𝑠 = 𝑡 → (𝐼𝑠) = (𝐼𝑡))
3 id 22 . . . . . 6 (𝑠 = 𝑡𝑠 = 𝑡)
42, 3sseq12d 3997 . . . . 5 (𝑠 = 𝑡 → ((𝐼𝑠) ⊆ 𝑠 ↔ (𝐼𝑡) ⊆ 𝑡))
54cbvralvw 3447 . . . 4 (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐼𝑡) ⊆ 𝑡)
65biimpi 217 . . 3 (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 → ∀𝑡 ∈ 𝒫 𝐵(𝐼𝑡) ⊆ 𝑡)
7 raaanv 4457 . . 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 4210 . . . . . . 7 (((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) → ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝑠𝑡))
109adantr 481 . . . . . 6 ((((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) ∧ (𝑠𝑡) = ∅) → ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝑠𝑡))
11 simpr 485 . . . . . 6 ((((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) ∧ (𝑠𝑡) = ∅) → (𝑠𝑡) = ∅)
1210, 11sseqtrd 4004 . . . . 5 ((((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) ∧ (𝑠𝑡) = ∅) → ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ ∅)
13 ss0 4349 . . . . 5 (((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)
1412, 13syl 17 . . . 4 ((((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) ∧ (𝑠𝑡) = ∅) → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)
1514ex 413 . . 3 (((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) → ((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅))
16152ralimi 3158 . 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 396   = wceq 1528  wral 3135  cin 3932  wss 3933  c0 4288  𝒫 cpw 4535  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356
This theorem is referenced by: (None)
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