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 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 4047	. 2 ⊢ ((𝐼‘∅) ∩ (𝐼‘∅)) = (𝐼‘∅)
2		0elpw 5056	. . 3 ⊢ ∅ ∈ 𝒫 𝐵
3		ineq1 4034	. . . . . . 7 ⊢ (𝑠 = ∅ → (𝑠 ∩ 𝑡) = (∅ ∩ 𝑡))
4	3	eqeq1d 2827	. . . . . 6 ⊢ (𝑠 = ∅ → ((𝑠 ∩ 𝑡) = ∅ ↔ (∅ ∩ 𝑡) = ∅))
5		fveq2 6433	. . . . . . . 8 ⊢ (𝑠 = ∅ → (𝐼‘𝑠) = (𝐼‘∅))
6	5	ineq1d 4040	. . . . . . 7 ⊢ (𝑠 = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ((𝐼‘∅) ∩ (𝐼‘𝑡)))
7	6	eqeq1d 2827	. . . . . 6 ⊢ (𝑠 = ∅ → (((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅ ↔ ((𝐼‘∅) ∩ (𝐼‘𝑡)) = ∅))
8	4, 7	imbi12d 336	. . . . 5 ⊢ (𝑠 = ∅ → (((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) ↔ ((∅ ∩ 𝑡) = ∅ → ((𝐼‘∅) ∩ (𝐼‘𝑡)) = ∅)))
9		0in 4194	. . . . . 6 ⊢ (∅ ∩ 𝑡) = ∅
10		pm5.5 353	. . . . . 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	syl6bb 279	. . . 4 ⊢ (𝑠 = ∅ → (((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) ↔ ((𝐼‘∅) ∩ (𝐼‘𝑡)) = ∅))
13		fveq2 6433	. . . . . 6 ⊢ (𝑡 = ∅ → (𝐼‘𝑡) = (𝐼‘∅))
14	13	ineq2d 4041	. . . . 5 ⊢ (𝑡 = ∅ → ((𝐼‘∅) ∩ (𝐼‘𝑡)) = ((𝐼‘∅) ∩ (𝐼‘∅)))
15	14	eqeq1d 2827	. . . 4 ⊢ (𝑡 = ∅ → (((𝐼‘∅) ∩ (𝐼‘𝑡)) = ∅ ↔ ((𝐼‘∅) ∩ (𝐼‘∅)) = ∅))
16	12, 15	rspc2v 3539	. . 3 ⊢ ((∅ ∈ 𝒫 𝐵 ∧ ∅ ∈ 𝒫 𝐵) → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) → ((𝐼‘∅) ∩ (𝐼‘∅)) = ∅))
17	2, 2, 16	mp2an 683	. 2 ⊢ (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) → ((𝐼‘∅) ∩ (𝐼‘∅)) = ∅)
18	1, 17	syl5eqr 2875	1 ⊢ (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) → (𝐼‘∅) = ∅)

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)