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Theorem disjss1 3751
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 2939 . . . . . 6  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21anim1d 319 . . . . 5  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  y  e.  C
)  ->  ( x  e.  B  /\  y  e.  C ) ) )
32alrimiv 1754 . . . 4  |-  ( A 
C_  B  ->  A. x
( ( x  e.  A  /\  y  e.  C )  ->  (
x  e.  B  /\  y  e.  C )
) )
4 moim 1964 . . . 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 14 . . 3  |-  ( A 
C_  B  ->  ( E* x ( x  e.  B  /\  y  e.  C )  ->  E* x ( x  e.  A  /\  y  e.  C ) ) )
65alimdv 1759 . 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 3747 . 2  |-  (Disj  x  e.  B  C  <->  A. y E* x ( x  e.  B  /\  y  e.  C ) )
8 dfdisj2 3747 . 2  |-  (Disj  x  e.  A  C  <->  A. y E* x ( x  e.  A  /\  y  e.  C ) )
96, 7, 83imtr4g 194 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 97   A.wal 1241    e. wcel 1393   E*wmo 1901    C_ wss 2917  Disj wdisj 3745
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-rmo 2314  df-in 2924  df-ss 2931  df-disj 3746
This theorem is referenced by:  disjeq1  3752  disjx0  3763
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