Theorem ntrk2imkb 41536

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 6756	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝐼‘𝑠) = (𝐼‘𝑡))
3		id 22	. . . . . 6 ⊢ (𝑠 = 𝑡 → 𝑠 = 𝑡)
4	2, 3	sseq12d 3950	. . . . 5 ⊢ (𝑠 = 𝑡 → ((𝐼‘𝑠) ⊆ 𝑠 ↔ (𝐼‘𝑡) ⊆ 𝑡))
5	4	cbvralvw 3372	. . . 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 4449	. . 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 582	. 2 ⊢ (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 → ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝐼‘𝑠) ⊆ 𝑠 ∧ (𝐼‘𝑡) ⊆ 𝑡))
9		ss2in 4167	. . . . . . 7 ⊢ (((𝐼‘𝑠) ⊆ 𝑠 ∧ (𝐼‘𝑡) ⊆ 𝑡) → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝑠 ∩ 𝑡))
10	9	adantr 480	. . . . . 6 ⊢ ((((𝐼‘𝑠) ⊆ 𝑠 ∧ (𝐼‘𝑡) ⊆ 𝑡) ∧ (𝑠 ∩ 𝑡) = ∅) → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝑠 ∩ 𝑡))
11		simpr 484	. . . . . 6 ⊢ ((((𝐼‘𝑠) ⊆ 𝑠 ∧ (𝐼‘𝑡) ⊆ 𝑡) ∧ (𝑠 ∩ 𝑡) = ∅) → (𝑠 ∩ 𝑡) = ∅)
12	10, 11	sseqtrd 3957	. . . . 5 ⊢ ((((𝐼‘𝑠) ⊆ 𝑠 ∧ (𝐼‘𝑡) ⊆ 𝑡) ∧ (𝑠 ∩ 𝑡) = ∅) → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ ∅)
13		ss0 4329	. . . . 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 412	. . 3 ⊢ (((𝐼‘𝑠) ⊆ 𝑠 ∧ (𝐼‘𝑡) ⊆ 𝑡) → ((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅))
16	15	2ralimi 3087	. 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 395   = wceq 1539  ∀wral 3063   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426

This theorem is referenced by: (None)