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Theorem ntrkbimka 37818
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 3800 . 2 ((𝐼‘∅) ∩ (𝐼‘∅)) = (𝐼‘∅)
2 0elpw 4794 . . 3 ∅ ∈ 𝒫 𝐵
3 ineq1 3785 . . . . . . 7 (𝑠 = ∅ → (𝑠𝑡) = (∅ ∩ 𝑡))
43eqeq1d 2623 . . . . . 6 (𝑠 = ∅ → ((𝑠𝑡) = ∅ ↔ (∅ ∩ 𝑡) = ∅))
5 fveq2 6148 . . . . . . . 8 (𝑠 = ∅ → (𝐼𝑠) = (𝐼‘∅))
65ineq1d 3791 . . . . . . 7 (𝑠 = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ((𝐼‘∅) ∩ (𝐼𝑡)))
76eqeq1d 2623 . . . . . 6 (𝑠 = ∅ → (((𝐼𝑠) ∩ (𝐼𝑡)) = ∅ ↔ ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅))
84, 7imbi12d 334 . . . . 5 (𝑠 = ∅ → (((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ↔ ((∅ ∩ 𝑡) = ∅ → ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅)))
9 0in 3941 . . . . . 6 (∅ ∩ 𝑡) = ∅
10 pm5.5 351 . . . . . 6 ((∅ ∩ 𝑡) = ∅ → (((∅ ∩ 𝑡) = ∅ → ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅))
119, 10ax-mp 5 . . . . 5 (((∅ ∩ 𝑡) = ∅ → ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅)
128, 11syl6bb 276 . . . 4 (𝑠 = ∅ → (((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐼‘∅) ∩ (𝐼𝑡)) = ∅))
13 fveq2 6148 . . . . . 6 (𝑡 = ∅ → (𝐼𝑡) = (𝐼‘∅))
1413ineq2d 3792 . . . . 5 (𝑡 = ∅ → ((𝐼‘∅) ∩ (𝐼𝑡)) = ((𝐼‘∅) ∩ (𝐼‘∅)))
1514eqeq1d 2623 . . . 4 (𝑡 = ∅ → (((𝐼‘∅) ∩ (𝐼𝑡)) = ∅ ↔ ((𝐼‘∅) ∩ (𝐼‘∅)) = ∅))
1612, 15rspc2v 3306 . . 3 ((∅ ∈ 𝒫 𝐵 ∧ ∅ ∈ 𝒫 𝐵) → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) → ((𝐼‘∅) ∩ (𝐼‘∅)) = ∅))
172, 2, 16mp2an 707 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) → ((𝐼‘∅) ∩ (𝐼‘∅)) = ∅)
181, 17syl5eqr 2669 1 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) → (𝐼‘∅) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1987  wral 2907  cin 3554  c0 3891  𝒫 cpw 4130  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855
This theorem is referenced by: (None)
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