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Theorem ntrkbimka 38336
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 |- ( A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) -> ( I ` (/) ) = (/) )
Distinct variable groups:   B, s, t   I, s, t
Proof of Theorem ntrkbimka
StepHypRef Expression
1 inidm 3822 . 2 |- ( ( I ` (/) ) i^i ( I ` (/) ) ) = ( I ` (/) )
2 0elpw 4834 . . 3 |- (/) e. ~P B
3 ineq1 3807 . . . . . . 7 |- ( s = (/) -> ( s i^i t ) = ( (/) i^i t ) )
43eqeq1d 2624 . . . . . 6 |- ( s = (/) -> ( ( s i^i t ) = (/) <-> ( (/) i^i t ) = (/) ) )
5 fveq2 6191 . . . . . . . 8 |- ( s = (/) -> ( I ` s ) = ( I ` (/) ) )
65ineq1d 3813 . . . . . . 7 |- ( s = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = ( ( I ` (/) ) i^i ( I ` t ) ) )
76eqeq1d 2624 . . . . . 6 |- ( s = (/) -> ( ( ( I ` s ) i^i ( I ` t ) ) = (/) <-> ( ( I ` (/) ) i^i ( I ` t ) ) = (/) ) )
84, 7imbi12d 334 . . . . 5 |- ( s = (/) -> ( ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) <-> ( ( (/) i^i t ) = (/) -> ( ( I ` (/) ) i^i ( I ` t ) ) = (/) ) ) )
9 0in 3969 . . . . . 6 |- ( (/) i^i t ) = (/)
10 pm5.5 351 . . . . . 6 |- ( ( (/) i^i t ) = (/) -> ( ( ( (/) i^i t ) = (/) -> ( ( I ` (/) ) i^i ( I ` t ) ) = (/) ) <-> ( ( I ` (/) ) i^i ( I ` t ) ) = (/) ) )
119, 10ax-mp 5 . . . . 5 |- ( ( ( (/) i^i t ) = (/) -> ( ( I ` (/) ) i^i ( I ` t ) ) = (/) ) <-> ( ( I ` (/) ) i^i ( I ` t ) ) = (/) )
128, 11syl6bb 276 . . . 4 |- ( s = (/) -> ( ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) <-> ( ( I ` (/) ) i^i ( I ` t ) ) = (/) ) )
13 fveq2 6191 . . . . . 6 |- ( t = (/) -> ( I ` t ) = ( I ` (/) ) )
1413ineq2d 3814 . . . . 5 |- ( t = (/) -> ( ( I ` (/) ) i^i ( I ` t ) ) = ( ( I ` (/) ) i^i ( I ` (/) ) ) )
1514eqeq1d 2624 . . . 4 |- ( t = (/) -> ( ( ( I ` (/) ) i^i ( I ` t ) ) = (/) <-> ( ( I ` (/) ) i^i ( I ` (/) ) ) = (/) ) )
1612, 15rspc2v 3322 . . 3 |- ( ( (/) e. ~P B /\ (/) e. ~P B ) -> ( A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) -> ( ( I ` (/) ) i^i ( I ` (/) ) ) = (/) ) )
172, 2, 16mp2an 708 . 2 |- ( A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) -> ( ( I ` (/) ) i^i ( I ` (/) ) ) = (/) )
181, 17syl5eqr 2670 1 |- ( A. s e. ~P B A. t e. ~P B ( ( s i^i t ) = (/) -> ( ( I ` s ) i^i ( I ` t ) ) = (/) ) -> ( I ` (/) ) = (/) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   (/)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  ax-nul 4789
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-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-pw 4160  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)
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