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Theorem disjss2 4420
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 3498 . . . . 5  |-  ( B 
C_  C  ->  (
y  e.  B  -> 
y  e.  C ) )
21ralimi 2857 . . . 4  |-  ( A. x  e.  A  B  C_  C  ->  A. x  e.  A  ( y  e.  B  ->  y  e.  C ) )
3 rmoim 3303 . . . 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 1685 . 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 4418 . 2  |-  (Disj  x  e.  A  C  <->  A. y E* x  e.  A  y  e.  C )
7 df-disj 4418 . 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 1377    e. wcel 1767   A.wral 2814   E*wrmo 2817    C_ wss 3476  Disj wdisj 4417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-ral 2819  df-rmo 2822  df-in 3483  df-ss 3490  df-disj 4418
This theorem is referenced by:  disjeq2  4421  0disj  4440  uniioombllem2  21727  uniioombllem4  21730  disjxwwlks  24412  disjxwwlkn  24421  usgreghash2spotv  24743
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