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Theorem iunss1 4048
Description: Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iunss1  |-  ( A 
C_  B  ->  U_ x  e.  A  C  C_  U_ x  e.  B  C )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem iunss1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssrexv 3353 . . 3  |-  ( A 
C_  B  ->  ( E. x  e.  A  y  e.  C  ->  E. x  e.  B  y  e.  C ) )
2 eliun 4041 . . 3  |-  ( y  e.  U_ x  e.  A  C  <->  E. x  e.  A  y  e.  C )
3 eliun 4041 . . 3  |-  ( y  e.  U_ x  e.  B  C  <->  E. x  e.  B  y  e.  C )
41, 2, 33imtr4g 262 . 2  |-  ( A 
C_  B  ->  (
y  e.  U_ x  e.  A  C  ->  y  e.  U_ x  e.  B  C ) )
54ssrdv 3299 1  |-  ( A 
C_  B  ->  U_ x  e.  A  C  C_  U_ x  e.  B  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   E.wrex 2652    C_ wss 3265   U_ciun 4037
This theorem is referenced by:  iuneq1  4050  iunxdif2  4082  oelim2  6776  fsumiun  12529  ovolfiniun  19266  uniioovol  19340  volsupnfl  25958  bnj1413  28744  bnj1408  28745
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-rex 2657  df-v 2903  df-in 3272  df-ss 3279  df-iun 4039
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