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Theorem disjss2 4371
Description: If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjss2  |-  ( A. x  e.  A  B  C_  C  ->  (Disj  x  e.  A  C  -> Disj  x  e.  A  B ) )

Proof of Theorem disjss2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssel 3438 . . . . 5  |-  ( B 
C_  C  ->  (
y  e.  B  -> 
y  e.  C ) )
21ralimi 2799 . . . 4  |-  ( A. x  e.  A  B  C_  C  ->  A. x  e.  A  ( y  e.  B  ->  y  e.  C ) )
3 rmoim 3251 . . . 4  |-  ( A. x  e.  A  (
y  e.  B  -> 
y  e.  C )  ->  ( E* x  e.  A  y  e.  C  ->  E* x  e.  A  y  e.  B
) )
42, 3syl 17 . . 3  |-  ( A. x  e.  A  B  C_  C  ->  ( E* x  e.  A  y  e.  C  ->  E* x  e.  A  y  e.  B ) )
54alimdv 1732 . 2  |-  ( A. x  e.  A  B  C_  C  ->  ( A. y E* x  e.  A  y  e.  C  ->  A. y E* x  e.  A  y  e.  B
) )
6 df-disj 4369 . 2  |-  (Disj  x  e.  A  C  <->  A. y E* x  e.  A  y  e.  C )
7 df-disj 4369 . 2  |-  (Disj  x  e.  A  B  <->  A. y E* x  e.  A  y  e.  B )
85, 6, 73imtr4g 272 1  |-  ( A. x  e.  A  B  C_  C  ->  (Disj  x  e.  A  C  -> Disj  x  e.  A  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1405    e. wcel 1844   A.wral 2756   E*wrmo 2759    C_ wss 3416  Disj wdisj 4368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-ral 2761  df-rmo 2764  df-in 3423  df-ss 3430  df-disj 4369
This theorem is referenced by:  disjeq2  4372  0disj  4390  uniioombllem2  22286  uniioombllem4  22289  disjxwwlks  25165  disjxwwlkn  25174  usgreghash2spotv  25495
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