Theorem ntrkbimka 41618

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 4158	. 2 ⊢ ((𝐼‘∅) ∩ (𝐼‘∅)) = (𝐼‘∅)
2		0elpw 5282	. . 3 ⊢ ∅ ∈ 𝒫 𝐵
3		ineq1 4145	. . . . . . 7 ⊢ (𝑠 = ∅ → (𝑠 ∩ 𝑡) = (∅ ∩ 𝑡))
4	3	eqeq1d 2742	. . . . . 6 ⊢ (𝑠 = ∅ → ((𝑠 ∩ 𝑡) = ∅ ↔ (∅ ∩ 𝑡) = ∅))
5		fveq2 6771	. . . . . . . 8 ⊢ (𝑠 = ∅ → (𝐼‘𝑠) = (𝐼‘∅))
6	5	ineq1d 4151	. . . . . . 7 ⊢ (𝑠 = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ((𝐼‘∅) ∩ (𝐼‘𝑡)))
7	6	eqeq1d 2742	. . . . . 6 ⊢ (𝑠 = ∅ → (((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅ ↔ ((𝐼‘∅) ∩ (𝐼‘𝑡)) = ∅))
8	4, 7	imbi12d 345	. . . . 5 ⊢ (𝑠 = ∅ → (((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) ↔ ((∅ ∩ 𝑡) = ∅ → ((𝐼‘∅) ∩ (𝐼‘𝑡)) = ∅)))
9		0in 4333	. . . . . 6 ⊢ (∅ ∩ 𝑡) = ∅
10		pm5.5 362	. . . . . 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 287	. . . 4 ⊢ (𝑠 = ∅ → (((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) ↔ ((𝐼‘∅) ∩ (𝐼‘𝑡)) = ∅))
13		fveq2 6771	. . . . . 6 ⊢ (𝑡 = ∅ → (𝐼‘𝑡) = (𝐼‘∅))
14	13	ineq2d 4152	. . . . 5 ⊢ (𝑡 = ∅ → ((𝐼‘∅) ∩ (𝐼‘𝑡)) = ((𝐼‘∅) ∩ (𝐼‘∅)))
15	14	eqeq1d 2742	. . . 4 ⊢ (𝑡 = ∅ → (((𝐼‘∅) ∩ (𝐼‘𝑡)) = ∅ ↔ ((𝐼‘∅) ∩ (𝐼‘∅)) = ∅))
16	12, 15	rspc2v 3571	. . 3 ⊢ ((∅ ∈ 𝒫 𝐵 ∧ ∅ ∈ 𝒫 𝐵) → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) → ((𝐼‘∅) ∩ (𝐼‘∅)) = ∅))
17	2, 2, 16	mp2an 689	. 2 ⊢ (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) → ((𝐼‘∅) ∩ (𝐼‘∅)) = ∅)
18	1, 17	eqtr3id 2794	1 ⊢ (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) → (𝐼‘∅) = ∅)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2110  ∀wral 3066   ∩ cin 3891  ∅c0 4262  𝒫 cpw 4539  ‘cfv 6432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-nul 5234

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-iota 6390  df-fv 6440

This theorem is referenced by: (None)