Theorem ntrkbimka 44841

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 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) → (𝐼‘∅) = ∅)

Distinct variable groups:   𝐵,𝑠,𝑡   𝐼,𝑠,𝑡

Proof of Theorem ntrkbimka

Step	Hyp	Ref	Expression
1		inidm 4179	. 2 ⊢ ((𝐼‘∅) ∩ (𝐼‘∅)) = (𝐼‘∅)
2		0elpw 5328	. . 3 ⊢ ∅ ∈ 𝒫 𝐵
3		ineq1 4166	. . . . . . 7 ⊢ (𝑠 = ∅ → (𝑠 ∩ 𝑡) = (∅ ∩ 𝑡))
4	3	eqeq1d 2767	. . . . . 6 ⊢ (𝑠 = ∅ → ((𝑠 ∩ 𝑡) = ∅ ↔ (∅ ∩ 𝑡) = ∅))
5		fveq2 6885	. . . . . . . 8 ⊢ (𝑠 = ∅ → (𝐼‘𝑠) = (𝐼‘∅))
6	5	ineq1d 4172	. . . . . . 7 ⊢ (𝑠 = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ((𝐼‘∅) ∩ (𝐼‘𝑡)))
7	6	eqeq1d 2767	. . . . . 6 ⊢ (𝑠 = ∅ → (((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅ ↔ ((𝐼‘∅) ∩ (𝐼‘𝑡)) = ∅))
8	4, 7	imbi12d 347	. . . . 5 ⊢ (𝑠 = ∅ → (((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) ↔ ((∅ ∩ 𝑡) = ∅ → ((𝐼‘∅) ∩ (𝐼‘𝑡)) = ∅)))
9		0in 4354	. . . . . 6 ⊢ (∅ ∩ 𝑡) = ∅
10		pm5.5 364	. . . . . 6 ⊢ ((∅ ∩ 𝑡) = ∅ → (((∅ ∩ 𝑡) = ∅ → ((𝐼‘∅) ∩ (𝐼‘𝑡)) = ∅) ↔ ((𝐼‘∅) ∩ (𝐼‘𝑡)) = ∅))
11	9, 10	ax-mp 5	. . . . 5 ⊢ (((∅ ∩ 𝑡) = ∅ → ((𝐼‘∅) ∩ (𝐼‘𝑡)) = ∅) ↔ ((𝐼‘∅) ∩ (𝐼‘𝑡)) = ∅)
12	8, 11	bitrdi 290	. . . 4 ⊢ (𝑠 = ∅ → (((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) ↔ ((𝐼‘∅) ∩ (𝐼‘𝑡)) = ∅))
13		fveq2 6885	. . . . . 6 ⊢ (𝑡 = ∅ → (𝐼‘𝑡) = (𝐼‘∅))
14	13	ineq2d 4173	. . . . 5 ⊢ (𝑡 = ∅ → ((𝐼‘∅) ∩ (𝐼‘𝑡)) = ((𝐼‘∅) ∩ (𝐼‘∅)))
15	14	eqeq1d 2767	. . . 4 ⊢ (𝑡 = ∅ → (((𝐼‘∅) ∩ (𝐼‘𝑡)) = ∅ ↔ ((𝐼‘∅) ∩ (𝐼‘∅)) = ∅))
16	12, 15	rspc2v 3594	. . 3 ⊢ ((∅ ∈ 𝒫 𝐵 ∧ ∅ ∈ 𝒫 𝐵) → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) → ((𝐼‘∅) ∩ (𝐼‘∅)) = ∅))
17	2, 2, 16	mp2an 705	. 2 ⊢ (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) → ((𝐼‘∅) ∩ (𝐼‘∅)) = ∅)
18	1, 17	eqtr3id 2814	1 ⊢ (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) → (𝐼‘∅) = ∅)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2146  ∀wral 3081   ∩ cin 3905  ∅c0 4286  𝒫 cpw 4564  ‘cfv 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-nul 5271

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548

This theorem is used by: (None)