Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  disjin Unicode version

Theorem disjin 23980
Description: If a collection is disjoint, so is the collection of the intersections with a given set. (Contributed by Thierry Arnoux, 14-Feb-2017.)
Assertion
Ref Expression
disjin  |-  (Disj  x  e.  B C  -> Disj  x  e.  B ( C  i^i  A ) )

Proof of Theorem disjin
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 inss1 3521 . . . . . . 7  |-  ( C  i^i  A )  C_  C
21sseli 3304 . . . . . 6  |-  ( y  e.  ( C  i^i  A )  ->  y  e.  C )
32anim2i 553 . . . . 5  |-  ( ( x  e.  B  /\  y  e.  ( C  i^i  A ) )  -> 
( x  e.  B  /\  y  e.  C
) )
43ax-gen 1552 . . . 4  |-  A. x
( ( x  e.  B  /\  y  e.  ( C  i^i  A
) )  ->  (
x  e.  B  /\  y  e.  C )
)
54rmoimi2 3095 . . 3  |-  ( E* x  e.  B y  e.  C  ->  E* x  e.  B y  e.  ( C  i^i  A
) )
65alimi 1565 . 2  |-  ( A. y E* x  e.  B
y  e.  C  ->  A. y E* x  e.  B y  e.  ( C  i^i  A ) )
7 df-disj 4143 . 2  |-  (Disj  x  e.  B C  <->  A. y E* x  e.  B
y  e.  C )
8 df-disj 4143 . 2  |-  (Disj  x  e.  B ( C  i^i  A )  <->  A. y E* x  e.  B y  e.  ( C  i^i  A ) )
96, 7, 83imtr4i 258 1  |-  (Disj  x  e.  B C  -> Disj  x  e.  B ( C  i^i  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546    e. wcel 1721   E*wrmo 2669    i^i cin 3279  Disj wdisj 4142
This theorem is referenced by:  measinblem  24527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rmo 2674  df-v 2918  df-in 3287  df-ss 3294  df-disj 4143
  Copyright terms: Public domain W3C validator