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Theorem ssiun 3772
Description: Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ssiun  |-  ( E. x  e.  A  C  C_  B  ->  C  C_  U_ x  e.  A  B )
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem ssiun
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssel 3019 . . . . 5  |-  ( C 
C_  B  ->  (
y  e.  C  -> 
y  e.  B ) )
21reximi 2470 . . . 4  |-  ( E. x  e.  A  C  C_  B  ->  E. x  e.  A  ( y  e.  C  ->  y  e.  B ) )
3 r19.37av 2520 . . . 4  |-  ( E. x  e.  A  ( y  e.  C  -> 
y  e.  B )  ->  ( y  e.  C  ->  E. x  e.  A  y  e.  B ) )
42, 3syl 14 . . 3  |-  ( E. x  e.  A  C  C_  B  ->  ( y  e.  C  ->  E. x  e.  A  y  e.  B ) )
5 eliun 3734 . . 3  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
64, 5syl6ibr 160 . 2  |-  ( E. x  e.  A  C  C_  B  ->  ( y  e.  C  ->  y  e. 
U_ x  e.  A  B ) )
76ssrdv 3031 1  |-  ( E. x  e.  A  C  C_  B  ->  C  C_  U_ x  e.  A  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1438   E.wrex 2360    C_ wss 2999   U_ciun 3730
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-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3005  df-ss 3012  df-iun 3732
This theorem is referenced by:  iunss2  3775  iunpwss  3820  iunpw  4302
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