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Theorem iunss1 3916
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 3238 . . 3  |-  ( A 
C_  B  ->  ( E. x  e.  A  y  e.  C  ->  E. x  e.  B  y  e.  C ) )
2 eliun 3909 . . 3  |-  ( y  e.  U_ x  e.  A  C  <->  E. x  e.  A  y  e.  C )
3 eliun 3909 . . 3  |-  ( y  e.  U_ x  e.  B  C  <->  E. x  e.  B  y  e.  C )
41, 2, 33imtr4g 261 . 2  |-  ( A 
C_  B  ->  (
y  e.  U_ x  e.  A  C  ->  y  e.  U_ x  e.  B  C ) )
54ssrdv 3185 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 1684   E.wrex 2544    C_ wss 3152   U_ciun 3905
This theorem is referenced by:  iuneq1  3918  iunxdif2  3950  oelim2  6593  fsumiun  12279  ovolfiniun  18860  uniioovol  18934  bnj1413  29065  bnj1408  29066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-iun 3907
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