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Theorem disjss1 3999
Description: A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjss1  |-  ( A 
C_  B  ->  (Disj  x  e.  B C  -> Disj  x  e.  A C ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem disjss1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssel 3174 . . . . . 6  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21anim1d 547 . . . . 5  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  y  e.  C
)  ->  ( x  e.  B  /\  y  e.  C ) ) )
32alrimiv 1617 . . . 4  |-  ( A 
C_  B  ->  A. x
( ( x  e.  A  /\  y  e.  C )  ->  (
x  e.  B  /\  y  e.  C )
) )
4 moim 2189 . . . 4  |-  ( A. x ( ( x  e.  A  /\  y  e.  C )  ->  (
x  e.  B  /\  y  e.  C )
)  ->  ( E* x ( x  e.  B  /\  y  e.  C )  ->  E* x ( x  e.  A  /\  y  e.  C ) ) )
53, 4syl 15 . . 3  |-  ( A 
C_  B  ->  ( E* x ( x  e.  B  /\  y  e.  C )  ->  E* x ( x  e.  A  /\  y  e.  C ) ) )
65alimdv 1607 . 2  |-  ( A 
C_  B  ->  ( A. y E* x ( x  e.  B  /\  y  e.  C )  ->  A. y E* x
( x  e.  A  /\  y  e.  C
) ) )
7 dfdisj2 3995 . 2  |-  (Disj  x  e.  B C  <->  A. y E* x ( x  e.  B  /\  y  e.  C ) )
8 dfdisj2 3995 . 2  |-  (Disj  x  e.  A C  <->  A. y E* x ( x  e.  A  /\  y  e.  C ) )
96, 7, 83imtr4g 261 1  |-  ( A 
C_  B  ->  (Disj  x  e.  B C  -> Disj  x  e.  A C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527    e. wcel 1684   E*wmo 2144    C_ wss 3152  Disj wdisj 3993
This theorem is referenced by:  disjeq1  4000  disjx0  4018  disjxiun  4020  disjss3  4022  volfiniun  18904  uniioovol  18934  uniioombllem4  18941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-rmo 2551  df-in 3159  df-ss 3166  df-disj 3994
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