Theorem ntrk2imkb 41647

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 22	. . 3 ⊢ (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 → ∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠)
2		fveq2 6774	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝐼‘𝑠) = (𝐼‘𝑡))
3		id 22	. . . . . 6 ⊢ (𝑠 = 𝑡 → 𝑠 = 𝑡)
4	2, 3	sseq12d 3954	. . . . 5 ⊢ (𝑠 = 𝑡 → ((𝐼‘𝑠) ⊆ 𝑠 ↔ (𝐼‘𝑡) ⊆ 𝑡))
5	4	cbvralvw 3383	. . . 4 ⊢ (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐼‘𝑡) ⊆ 𝑡)
6	5	biimpi 215	. . 3 ⊢ (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 → ∀𝑡 ∈ 𝒫 𝐵(𝐼‘𝑡) ⊆ 𝑡)
7		raaanv 4452	. . 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 583	. 2 ⊢ (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 → ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝐼‘𝑠) ⊆ 𝑠 ∧ (𝐼‘𝑡) ⊆ 𝑡))
9		ss2in 4170	. . . . . . 7 ⊢ (((𝐼‘𝑠) ⊆ 𝑠 ∧ (𝐼‘𝑡) ⊆ 𝑡) → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝑠 ∩ 𝑡))
10	9	adantr 481	. . . . . 6 ⊢ ((((𝐼‘𝑠) ⊆ 𝑠 ∧ (𝐼‘𝑡) ⊆ 𝑡) ∧ (𝑠 ∩ 𝑡) = ∅) → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝑠 ∩ 𝑡))
11		simpr 485	. . . . . 6 ⊢ ((((𝐼‘𝑠) ⊆ 𝑠 ∧ (𝐼‘𝑡) ⊆ 𝑡) ∧ (𝑠 ∩ 𝑡) = ∅) → (𝑠 ∩ 𝑡) = ∅)
12	10, 11	sseqtrd 3961	. . . . 5 ⊢ ((((𝐼‘𝑠) ⊆ 𝑠 ∧ (𝐼‘𝑡) ⊆ 𝑡) ∧ (𝑠 ∩ 𝑡) = ∅) → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ ∅)
13		ss0 4332	. . . . 5 ⊢ (((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅)
14	12, 13	syl 17	. . . 4 ⊢ ((((𝐼‘𝑠) ⊆ 𝑠 ∧ (𝐼‘𝑡) ⊆ 𝑡) ∧ (𝑠 ∩ 𝑡) = ∅) → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅)
15	14	ex 413	. . 3 ⊢ (((𝐼‘𝑠) ⊆ 𝑠 ∧ (𝐼‘𝑡) ⊆ 𝑡) → ((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅))
16	15	2ralimi 3088	. 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 17	1 ⊢ (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 → ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539  ∀wral 3064   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441

This theorem is referenced by: (None)