Theorem ntrk2imkb 44796

Description: If an interior function is contracting, the interiors of disjoint sets are disjoint. Kuratowski's K2 axiom implies KB. Interior version. (Contributed by RP, 9-Jun-2021.)

Assertion

Ref	Expression
ntrk2imkb	⊢ (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 → ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅))

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

Proof of Theorem ntrk2imkb

Step	Hyp	Ref	Expression
1		id 23	. . 3 ⊢ (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 → ∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠)
2		fveq2 6885	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝐼‘𝑠) = (𝐼‘𝑡))
3		id 23	. . . . . 6 ⊢ (𝑠 = 𝑡 → 𝑠 = 𝑡)
4	2, 3	sseq12d 3971	. . . . 5 ⊢ (𝑠 = 𝑡 → ((𝐼‘𝑠) ⊆ 𝑠 ↔ (𝐼‘𝑡) ⊆ 𝑡))
5	4	cbvralvw 3245	. . . 4 ⊢ (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐼‘𝑡) ⊆ 𝑡)
6	5	biimpi 219	. . 3 ⊢ (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 → ∀𝑡 ∈ 𝒫 𝐵(𝐼‘𝑡) ⊆ 𝑡)
7		raaanv 4482	. . 3 ⊢ (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝐼‘𝑠) ⊆ 𝑠 ∧ (𝐼‘𝑡) ⊆ 𝑡) ↔ (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 ∧ ∀𝑡 ∈ 𝒫 𝐵(𝐼‘𝑡) ⊆ 𝑡))
8	1, 6, 7	sylanbrc 595	. 2 ⊢ (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 → ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝐼‘𝑠) ⊆ 𝑠 ∧ (𝐼‘𝑡) ⊆ 𝑡))
9		ss2in 4197	. . . . . . 7 ⊢ (((𝐼‘𝑠) ⊆ 𝑠 ∧ (𝐼‘𝑡) ⊆ 𝑡) → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝑠 ∩ 𝑡))
10	9	adantr 486	. . . . . 6 ⊢ ((((𝐼‘𝑠) ⊆ 𝑠 ∧ (𝐼‘𝑡) ⊆ 𝑡) ∧ (𝑠 ∩ 𝑡) = ∅) → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝑠 ∩ 𝑡))
11		simpr 490	. . . . . 6 ⊢ ((((𝐼‘𝑠) ⊆ 𝑠 ∧ (𝐼‘𝑡) ⊆ 𝑡) ∧ (𝑠 ∩ 𝑡) = ∅) → (𝑠 ∩ 𝑡) = ∅)
12	10, 11	sseqtrd 3974	. . . . 5 ⊢ ((((𝐼‘𝑠) ⊆ 𝑠 ∧ (𝐼‘𝑡) ⊆ 𝑡) ∧ (𝑠 ∩ 𝑡) = ∅) → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ ∅)
13		ss0 4359	. . . . 5 ⊢ (((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅)
14	12, 13	syl 18	. . . 4 ⊢ ((((𝐼‘𝑠) ⊆ 𝑠 ∧ (𝐼‘𝑡) ⊆ 𝑡) ∧ (𝑠 ∩ 𝑡) = ∅) → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅)
15	14	ex 418	. . 3 ⊢ (((𝐼‘𝑠) ⊆ 𝑠 ∧ (𝐼‘𝑡) ⊆ 𝑡) → ((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅))
16	15	2ralimi 3137	. 2 ⊢ (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝐼‘𝑠) ⊆ 𝑠 ∧ (𝐼‘𝑡) ⊆ 𝑡) → ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅))
17	8, 16	syl 18	1 ⊢ (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 → ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∀wral 3081   ∩ cin 3905   ⊆ wss 3906  ∅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-12 2216  ax-ext 2737

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-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  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-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)