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Theorem ntrk2imkb 38861
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 6332 . . . . . 6 (𝑠 = 𝑡 → (𝐼𝑠) = (𝐼𝑡))
3 id 22 . . . . . 6 (𝑠 = 𝑡𝑠 = 𝑡)
42, 3sseq12d 3783 . . . . 5 (𝑠 = 𝑡 → ((𝐼𝑠) ⊆ 𝑠 ↔ (𝐼𝑡) ⊆ 𝑡))
54cbvralv 3320 . . . 4 (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐼𝑡) ⊆ 𝑡)
65biimpi 206 . . 3 (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 → ∀𝑡 ∈ 𝒫 𝐵(𝐼𝑡) ⊆ 𝑡)
7 raaanv 4222 . . 3 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) ↔ (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 ∧ ∀𝑡 ∈ 𝒫 𝐵(𝐼𝑡) ⊆ 𝑡))
81, 6, 7sylanbrc 572 . 2 (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 → ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡))
9 ss2in 3989 . . . . . . 7 (((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) → ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝑠𝑡))
109adantr 466 . . . . . 6 ((((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) ∧ (𝑠𝑡) = ∅) → ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝑠𝑡))
11 simpr 471 . . . . . 6 ((((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) ∧ (𝑠𝑡) = ∅) → (𝑠𝑡) = ∅)
1210, 11sseqtrd 3790 . . . . 5 ((((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) ∧ (𝑠𝑡) = ∅) → ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ ∅)
13 ss0 4118 . . . . 5 (((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)
1412, 13syl 17 . . . 4 ((((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) ∧ (𝑠𝑡) = ∅) → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)
1514ex 397 . . 3 (((𝐼𝑠) ⊆ 𝑠 ∧ (𝐼𝑡) ⊆ 𝑡) → ((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅))
16152ralimi 3102 . 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 382   = wceq 1631  wral 3061  cin 3722  wss 3723  c0 4063  𝒫 cpw 4297  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039
This theorem is referenced by: (None)
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