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Theorem disjss2 4260
Description: If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjss2  |-  ( A. x  e.  A  B  C_  C  ->  (Disj  x  e.  A  C  -> Disj  x  e.  A  B ) )

Proof of Theorem disjss2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssel 3345 . . . . 5  |-  ( B 
C_  C  ->  (
y  e.  B  -> 
y  e.  C ) )
21ralimi 2786 . . . 4  |-  ( A. x  e.  A  B  C_  C  ->  A. x  e.  A  ( y  e.  B  ->  y  e.  C ) )
3 rmoim 3153 . . . 4  |-  ( A. x  e.  A  (
y  e.  B  -> 
y  e.  C )  ->  ( E* x  e.  A  y  e.  C  ->  E* x  e.  A  y  e.  B
) )
42, 3syl 16 . . 3  |-  ( A. x  e.  A  B  C_  C  ->  ( E* x  e.  A  y  e.  C  ->  E* x  e.  A  y  e.  B ) )
54alimdv 1675 . 2  |-  ( A. x  e.  A  B  C_  C  ->  ( A. y E* x  e.  A  y  e.  C  ->  A. y E* x  e.  A  y  e.  B
) )
6 df-disj 4258 . 2  |-  (Disj  x  e.  A  C  <->  A. y E* x  e.  A  y  e.  C )
7 df-disj 4258 . 2  |-  (Disj  x  e.  A  B  <->  A. y E* x  e.  A  y  e.  B )
85, 6, 73imtr4g 270 1  |-  ( A. x  e.  A  B  C_  C  ->  (Disj  x  e.  A  C  -> Disj  x  e.  A  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1367    e. wcel 1756   A.wral 2710   E*wrmo 2713    C_ wss 3323  Disj wdisj 4257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-ral 2715  df-rmo 2718  df-in 3330  df-ss 3337  df-disj 4258
This theorem is referenced by:  disjeq2  4261  0disj  4280  uniioombllem2  21038  uniioombllem4  21041  disjxwwlks  30321  disjxwwlkn  30517  usgreghash2spotv  30612
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