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Theorem disjss1 4371
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 3435 . . . . . 6  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21anim1d 562 . . . . 5  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  y  e.  C
)  ->  ( x  e.  B  /\  y  e.  C ) ) )
32alrimiv 1740 . . . 4  |-  ( A 
C_  B  ->  A. x
( ( x  e.  A  /\  y  e.  C )  ->  (
x  e.  B  /\  y  e.  C )
) )
4 moim 2291 . . . 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 17 . . 3  |-  ( A 
C_  B  ->  ( E* x ( x  e.  B  /\  y  e.  C )  ->  E* x ( x  e.  A  /\  y  e.  C ) ) )
65alimdv 1730 . 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 4367 . 2  |-  (Disj  x  e.  B  C  <->  A. y E* x ( x  e.  B  /\  y  e.  C ) )
8 dfdisj2 4367 . 2  |-  (Disj  x  e.  A  C  <->  A. y E* x ( x  e.  A  /\  y  e.  C ) )
96, 7, 83imtr4g 270 1  |-  ( A 
C_  B  ->  (Disj  x  e.  B  C  -> Disj  x  e.  A  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wal 1403    e. wcel 1842   E*wmo 2239    C_ wss 3413  Disj wdisj 4365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-rmo 2761  df-in 3420  df-ss 3427  df-disj 4366
This theorem is referenced by:  disjeq1  4372  disjx0  4389  disjxiun  4391  disjss3  4393  volfiniun  22141  uniioovol  22172  uniioombllem4  22179  disjiunel  27768  carsggect  28646  carsgclctunlem2  28647  omsmeas  28651  sibfof  28668
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