MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  disjss1 Structured version   Unicode version

Theorem disjss1 4190
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 3344 . . . . . 6  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21anim1d 549 . . . . 5  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  y  e.  C
)  ->  ( x  e.  B  /\  y  e.  C ) ) )
32alrimiv 1642 . . . 4  |-  ( A 
C_  B  ->  A. x
( ( x  e.  A  /\  y  e.  C )  ->  (
x  e.  B  /\  y  e.  C )
) )
4 moim 2329 . . . 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 1632 . 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 4186 . 2  |-  (Disj  x  e.  B C  <->  A. y E* x ( x  e.  B  /\  y  e.  C ) )
8 dfdisj2 4186 . 2  |-  (Disj  x  e.  A C  <->  A. y E* x ( x  e.  A  /\  y  e.  C ) )
96, 7, 83imtr4g 263 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 360   A.wal 1550    e. wcel 1726   E*wmo 2284    C_ wss 3322  Disj wdisj 4184
This theorem is referenced by:  disjeq1  4191  disjx0  4209  disjxiun  4211  disjss3  4213  volfiniun  19443  uniioovol  19473  uniioombllem4  19480  sibfof  24656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-rmo 2715  df-in 3329  df-ss 3336  df-disj 4185
  Copyright terms: Public domain W3C validator