Theorem ntrk2imkb 38335

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	|- ( A. s e. ~P B ( I ` s ) C_ s -> A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) )

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

Proof of Theorem ntrk2imkb

Step	Hyp	Ref	Expression
1		id 22	. . 3 |- ( A. s e. ~P B ( I ` s ) C_ s -> A. s e. ~P B ( I ` s ) C_ s )
2		fveq2 6191	. . . . . 6 |- ( s = t -> ( I ` s ) = ( I ` t ) )
3		id 22	. . . . . 6 |- ( s = t -> s = t )
4	2, 3	sseq12d 3634	. . . . 5 |- ( s = t -> ( ( I ` s ) C_ s <-> ( I ` t ) C_ t ) )
5	4	cbvralv 3171	. . . 4 |- ( A. s e. ~P B ( I ` s ) C_ s <-> A. t e. ~P B ( I ` t ) C_ t )
6	5	biimpi 206	. . 3 |- ( A. s e. ~P B ( I ` s ) C_ s -> A. t e. ~P B ( I ` t ) C_ t )
7		raaanv 4083	. . 3 |- ( A. s e. ~P B A. t e. ~P B ( ( I ` s ) C_ s /\ ( I ` t ) C_ t ) <-> ( A. s e. ~P B ( I ` s ) C_ s /\ A. t e. ~P B ( I ` t ) C_ t ) )
8	1, 6, 7	sylanbrc 698	. 2 |- ( A. s e. ~P B ( I ` s ) C_ s -> A. s e. ~P B A. t e. ~P B ( ( I ` s ) C_ s /\ ( I ` t ) C_ t ) )
9		ss2in 3840	. . . . . . 7 |- ( ( ( I ` s ) C_ s /\ ( I ` t ) C_ t ) -> ( ( I ` s ) i^i ( I ` t ) ) C_ ( s i^i t ) )
10	9	adantr 481	. . . . . 6 |- ( ( ( ( I ` s ) C_ s /\ ( I ` t ) C_ t ) /\ ( s i^i t ) = (/) ) -> ( ( I ` s ) i^i ( I ` t ) ) C_ ( s i^i t ) )
11		simpr 477	. . . . . 6 |- ( ( ( ( I ` s ) C_ s /\ ( I ` t ) C_ t ) /\ ( s i^i t ) = (/) ) -> ( s i^i t ) = (/) )
12	10, 11	sseqtrd 3641	. . . . 5 |- ( ( ( ( I ` s ) C_ s /\ ( I ` t ) C_ t ) /\ ( s i^i t ) = (/) ) -> ( ( I ` s ) i^i ( I ` t ) ) C_ (/) )
13		ss0 3974	. . . . 5 |- ( ( ( I ` s ) i^i ( I ` t ) ) C_ (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) )
14	12, 13	syl 17	. . . 4 |- ( ( ( ( I ` s ) C_ s /\ ( I ` t ) C_ t ) /\ ( s i^i t ) = (/) ) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) )
15	14	ex 450	. . 3 |- ( ( ( I ` s ) C_ s /\ ( I ` t ) C_ t ) -> ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) )
16	15	2ralimi 2953	. 2 |- ( A. s e. ~P B A. t e. ~P B ( ( I ` s ) C_ s /\ ( I ` t ) C_ t ) -> A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) )
17	8, 16	syl 17	1 |- ( A. s e. ~P B ( I ` s ) C_ s -> A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   A.wral 2912   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by: (None)