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Theorem disjss2 3947
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 3122 . . . . 5  |-  ( B 
C_  C  ->  (
y  e.  B  -> 
y  e.  C ) )
21ralimi 2520 . . . 4  |-  ( A. x  e.  A  B  C_  C  ->  A. x  e.  A  ( y  e.  B  ->  y  e.  C ) )
3 rmoim 2913 . . . 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 14 . . 3  |-  ( A. x  e.  A  B  C_  C  ->  ( E* x  e.  A  y  e.  C  ->  E* x  e.  A  y  e.  B ) )
54alimdv 1859 . 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 3945 . 2  |-  (Disj  x  e.  A  C  <->  A. y E* x  e.  A  y  e.  C )
7 df-disj 3945 . 2  |-  (Disj  x  e.  A  B  <->  A. y E* x  e.  A  y  e.  B )
85, 6, 73imtr4g 204 1  |-  ( A. x  e.  A  B  C_  C  ->  (Disj  x  e.  A  C  -> Disj  x  e.  A  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1333    e. wcel 2128   A.wral 2435   E*wrmo 2438    C_ wss 3102  Disj wdisj 3944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-ral 2440  df-rmo 2443  df-in 3108  df-ss 3115  df-disj 3945
This theorem is referenced by:  disjeq2  3948  0disj  3964
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