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Theorem disjss2 3817
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 3017 . . . . 5  |-  ( B 
C_  C  ->  (
y  e.  B  -> 
y  e.  C ) )
21ralimi 2438 . . . 4  |-  ( A. x  e.  A  B  C_  C  ->  A. x  e.  A  ( y  e.  B  ->  y  e.  C ) )
3 rmoim 2814 . . . 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 1807 . 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 3815 . 2  |-  (Disj  x  e.  A  C  <->  A. y E* x  e.  A  y  e.  C )
7 df-disj 3815 . 2  |-  (Disj  x  e.  A  B  <->  A. y E* x  e.  A  y  e.  B )
85, 6, 73imtr4g 203 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 1287    e. wcel 1438   A.wral 2359   E*wrmo 2362    C_ wss 2997  Disj wdisj 3814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-ral 2364  df-rmo 2367  df-in 3003  df-ss 3010  df-disj 3815
This theorem is referenced by:  disjeq2  3818  0disj  3834
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