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Theorem ntrkbimka 42398
Description: If the interiors of disjoint sets are disjoint, then the interior of the empty set is the empty set. (Contributed by RP, 14-Jun-2021.)
Assertion
Ref Expression
ntrkbimka (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) → (𝐼‘∅) = ∅)
Distinct variable groups:   𝐵,𝑠,𝑡   𝐼,𝑠,𝑡

Proof of Theorem ntrkbimka
StepHypRef Expression
1 inidm 4179 . 2 ((𝐼‘∅) ∩ (𝐼‘∅)) = (𝐼‘∅)
2 0elpw 5312 . . 3 ∅ ∈ 𝒫 𝐵
3 ineq1 4166 . . . . . . 7 (𝑠 = ∅ → (𝑠𝑡) = (∅ ∩ 𝑡))
43eqeq1d 2735 . . . . . 6 (𝑠 = ∅ → ((𝑠𝑡) = ∅ ↔ (∅ ∩ 𝑡) = ∅))
5 fveq2 6843 . . . . . . . 8 (𝑠 = ∅ → (𝐼𝑠) = (𝐼‘∅))
65ineq1d 4172 . . . . . . 7 (𝑠 = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ((𝐼‘∅) ∩ (𝐼𝑡)))
76eqeq1d 2735 . . . . . 6 (𝑠 = ∅ → (((𝐼𝑠) ∩ (𝐼𝑡)) = ∅ ↔ ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅))
84, 7imbi12d 345 . . . . 5 (𝑠 = ∅ → (((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ↔ ((∅ ∩ 𝑡) = ∅ → ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅)))
9 0in 4354 . . . . . 6 (∅ ∩ 𝑡) = ∅
10 pm5.5 362 . . . . . 6 ((∅ ∩ 𝑡) = ∅ → (((∅ ∩ 𝑡) = ∅ → ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅))
119, 10ax-mp 5 . . . . 5 (((∅ ∩ 𝑡) = ∅ → ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅)
128, 11bitrdi 287 . . . 4 (𝑠 = ∅ → (((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅))
13 fveq2 6843 . . . . . 6 (𝑡 = ∅ → (𝐼𝑡) = (𝐼‘∅))
1413ineq2d 4173 . . . . 5 (𝑡 = ∅ → ((𝐼‘∅) ∩ (𝐼𝑡)) = ((𝐼‘∅) ∩ (𝐼‘∅)))
1514eqeq1d 2735 . . . 4 (𝑡 = ∅ → (((𝐼‘∅) ∩ (𝐼𝑡)) = ∅ ↔ ((𝐼‘∅) ∩ (𝐼‘∅)) = ∅))
1612, 15rspc2v 3589 . . 3 ((∅ ∈ 𝒫 𝐵 ∧ ∅ ∈ 𝒫 𝐵) → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) → ((𝐼‘∅) ∩ (𝐼‘∅)) = ∅))
172, 2, 16mp2an 691 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) → ((𝐼‘∅) ∩ (𝐼‘∅)) = ∅)
181, 17eqtr3id 2787 1 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) → (𝐼‘∅) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  wral 3061  cin 3910  c0 4283  𝒫 cpw 4561  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5264
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505
This theorem is referenced by: (None)
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