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Theorem ntrkbimka 40555
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 4170 . 2 ((𝐼‘∅) ∩ (𝐼‘∅)) = (𝐼‘∅)
2 0elpw 5229 . . 3 ∅ ∈ 𝒫 𝐵
3 ineq1 4156 . . . . . . 7 (𝑠 = ∅ → (𝑠𝑡) = (∅ ∩ 𝑡))
43eqeq1d 2823 . . . . . 6 (𝑠 = ∅ → ((𝑠𝑡) = ∅ ↔ (∅ ∩ 𝑡) = ∅))
5 fveq2 6643 . . . . . . . 8 (𝑠 = ∅ → (𝐼𝑠) = (𝐼‘∅))
65ineq1d 4163 . . . . . . 7 (𝑠 = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ((𝐼‘∅) ∩ (𝐼𝑡)))
76eqeq1d 2823 . . . . . 6 (𝑠 = ∅ → (((𝐼𝑠) ∩ (𝐼𝑡)) = ∅ ↔ ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅))
84, 7imbi12d 348 . . . . 5 (𝑠 = ∅ → (((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ↔ ((∅ ∩ 𝑡) = ∅ → ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅)))
9 0in 4320 . . . . . 6 (∅ ∩ 𝑡) = ∅
10 pm5.5 365 . . . . . 6 ((∅ ∩ 𝑡) = ∅ → (((∅ ∩ 𝑡) = ∅ → ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅))
119, 10ax-mp 5 . . . . 5 (((∅ ∩ 𝑡) = ∅ → ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅)
128, 11syl6bb 290 . . . 4 (𝑠 = ∅ → (((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅))
13 fveq2 6643 . . . . . 6 (𝑡 = ∅ → (𝐼𝑡) = (𝐼‘∅))
1413ineq2d 4164 . . . . 5 (𝑡 = ∅ → ((𝐼‘∅) ∩ (𝐼𝑡)) = ((𝐼‘∅) ∩ (𝐼‘∅)))
1514eqeq1d 2823 . . . 4 (𝑡 = ∅ → (((𝐼‘∅) ∩ (𝐼𝑡)) = ∅ ↔ ((𝐼‘∅) ∩ (𝐼‘∅)) = ∅))
1612, 15rspc2v 3610 . . 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, 17syl5eqr 2870 1 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) → (𝐼‘∅) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2115  wral 3126  cin 3909  c0 4266  𝒫 cpw 4512  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-nul 5183
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-iota 6287  df-fv 6336
This theorem is referenced by: (None)
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