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Theorem ntrkbimka 39169
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 4047 . 2 ((𝐼‘∅) ∩ (𝐼‘∅)) = (𝐼‘∅)
2 0elpw 5056 . . 3 ∅ ∈ 𝒫 𝐵
3 ineq1 4034 . . . . . . 7 (𝑠 = ∅ → (𝑠𝑡) = (∅ ∩ 𝑡))
43eqeq1d 2827 . . . . . 6 (𝑠 = ∅ → ((𝑠𝑡) = ∅ ↔ (∅ ∩ 𝑡) = ∅))
5 fveq2 6433 . . . . . . . 8 (𝑠 = ∅ → (𝐼𝑠) = (𝐼‘∅))
65ineq1d 4040 . . . . . . 7 (𝑠 = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ((𝐼‘∅) ∩ (𝐼𝑡)))
76eqeq1d 2827 . . . . . 6 (𝑠 = ∅ → (((𝐼𝑠) ∩ (𝐼𝑡)) = ∅ ↔ ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅))
84, 7imbi12d 336 . . . . 5 (𝑠 = ∅ → (((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ↔ ((∅ ∩ 𝑡) = ∅ → ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅)))
9 0in 4194 . . . . . 6 (∅ ∩ 𝑡) = ∅
10 pm5.5 353 . . . . . 6 ((∅ ∩ 𝑡) = ∅ → (((∅ ∩ 𝑡) = ∅ → ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅))
119, 10ax-mp 5 . . . . 5 (((∅ ∩ 𝑡) = ∅ → ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅)
128, 11syl6bb 279 . . . 4 (𝑠 = ∅ → (((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅))
13 fveq2 6433 . . . . . 6 (𝑡 = ∅ → (𝐼𝑡) = (𝐼‘∅))
1413ineq2d 4041 . . . . 5 (𝑡 = ∅ → ((𝐼‘∅) ∩ (𝐼𝑡)) = ((𝐼‘∅) ∩ (𝐼‘∅)))
1514eqeq1d 2827 . . . 4 (𝑡 = ∅ → (((𝐼‘∅) ∩ (𝐼𝑡)) = ∅ ↔ ((𝐼‘∅) ∩ (𝐼‘∅)) = ∅))
1612, 15rspc2v 3539 . . 3 ((∅ ∈ 𝒫 𝐵 ∧ ∅ ∈ 𝒫 𝐵) → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) → ((𝐼‘∅) ∩ (𝐼‘∅)) = ∅))
172, 2, 16mp2an 683 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) → ((𝐼‘∅) ∩ (𝐼‘∅)) = ∅)
181, 17syl5eqr 2875 1 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) → (𝐼‘∅) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1656  wcel 2164  wral 3117  cin 3797  c0 4144  𝒫 cpw 4378  cfv 6123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-nul 5013
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-iota 6086  df-fv 6131
This theorem is referenced by: (None)
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