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Theorem disjss1f 23969
Description: A subset of a disjoint collection is disjoint. (Contributed by Thierry Arnoux, 6-Apr-2017.)
Hypotheses
Ref Expression
disjss1f.1  |-  F/_ x A
disjss1f.2  |-  F/_ x B
Assertion
Ref Expression
disjss1f  |-  ( A 
C_  B  ->  (Disj  x  e.  B C  -> Disj  x  e.  A C ) )

Proof of Theorem disjss1f
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 disjss1f.1 . . . 4  |-  F/_ x A
2 disjss1f.2 . . . 4  |-  F/_ x B
31, 2ssrmo 23934 . . 3  |-  ( A 
C_  B  ->  ( E* x  e.  B
y  e.  C  ->  E* x  e.  A
y  e.  C ) )
43alimdv 1628 . 2  |-  ( A 
C_  B  ->  ( A. y E* x  e.  B y  e.  C  ->  A. y E* x  e.  A y  e.  C
) )
5 df-disj 4143 . 2  |-  (Disj  x  e.  B C  <->  A. y E* x  e.  B
y  e.  C )
6 df-disj 4143 . 2  |-  (Disj  x  e.  A C  <->  A. y E* x  e.  A
y  e.  C )
74, 5, 63imtr4g 262 1  |-  ( A 
C_  B  ->  (Disj  x  e.  B C  -> Disj  x  e.  A C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1546    e. wcel 1721   F/_wnfc 2527   E*wrmo 2669    C_ wss 3280  Disj wdisj 4142
This theorem is referenced by:  measvuni  24521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rmo 2674  df-in 3287  df-ss 3294  df-disj 4143
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