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Theorem disjss1 4256
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 3338 . . . . . 6  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21anim1d 559 . . . . 5  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  y  e.  C
)  ->  ( x  e.  B  /\  y  e.  C ) ) )
32alrimiv 1684 . . . 4  |-  ( A 
C_  B  ->  A. x
( ( x  e.  A  /\  y  e.  C )  ->  (
x  e.  B  /\  y  e.  C )
) )
4 moim 2316 . . . 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 16 . . 3  |-  ( A 
C_  B  ->  ( E* x ( x  e.  B  /\  y  e.  C )  ->  E* x ( x  e.  A  /\  y  e.  C ) ) )
65alimdv 1674 . 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 4252 . 2  |-  (Disj  x  e.  B  C  <->  A. y E* x ( x  e.  B  /\  y  e.  C ) )
8 dfdisj2 4252 . 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 369   A.wal 1360    e. wcel 1755   E*wmo 2255    C_ wss 3316  Disj wdisj 4250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-rmo 2713  df-in 3323  df-ss 3330  df-disj 4251
This theorem is referenced by:  disjeq1  4257  disjx0  4275  disjxiun  4277  disjss3  4279  volfiniun  20869  uniioovol  20900  uniioombllem4  20907  sibfof  26573
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