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Theorem ntrkbimka 43941
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 4242 . 2 ((𝐼‘∅) ∩ (𝐼‘∅)) = (𝐼‘∅)
2 0elpw 5377 . . 3 ∅ ∈ 𝒫 𝐵
3 ineq1 4228 . . . . . . 7 (𝑠 = ∅ → (𝑠𝑡) = (∅ ∩ 𝑡))
43eqeq1d 2736 . . . . . 6 (𝑠 = ∅ → ((𝑠𝑡) = ∅ ↔ (∅ ∩ 𝑡) = ∅))
5 fveq2 6919 . . . . . . . 8 (𝑠 = ∅ → (𝐼𝑠) = (𝐼‘∅))
65ineq1d 4234 . . . . . . 7 (𝑠 = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ((𝐼‘∅) ∩ (𝐼𝑡)))
76eqeq1d 2736 . . . . . 6 (𝑠 = ∅ → (((𝐼𝑠) ∩ (𝐼𝑡)) = ∅ ↔ ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅))
84, 7imbi12d 344 . . . . 5 (𝑠 = ∅ → (((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ↔ ((∅ ∩ 𝑡) = ∅ → ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅)))
9 0in 4416 . . . . . 6 (∅ ∩ 𝑡) = ∅
10 pm5.5 361 . . . . . 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 6919 . . . . . 6 (𝑡 = ∅ → (𝐼𝑡) = (𝐼‘∅))
1413ineq2d 4235 . . . . 5 (𝑡 = ∅ → ((𝐼‘∅) ∩ (𝐼𝑡)) = ((𝐼‘∅) ∩ (𝐼‘∅)))
1514eqeq1d 2736 . . . 4 (𝑡 = ∅ → (((𝐼‘∅) ∩ (𝐼𝑡)) = ∅ ↔ ((𝐼‘∅) ∩ (𝐼‘∅)) = ∅))
1612, 15rspc2v 3642 . . 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 2788 1 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) → (𝐼‘∅) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2103  wral 3063  cin 3969  c0 4347  𝒫 cpw 4622  cfv 6572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705  ax-nul 5327
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3064  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-iota 6524  df-fv 6580
This theorem is referenced by: (None)
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