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Theorem ssiun 3943
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 3164 . . . . 5  |-  ( C 
C_  B  ->  (
y  e.  C  -> 
y  e.  B ) )
21reximi 2587 . . . 4  |-  ( E. x  e.  A  C  C_  B  ->  E. x  e.  A  ( y  e.  C  ->  y  e.  B ) )
3 r19.37av 2643 . . . 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 3905 . . 3  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
64, 5imbitrrdi 162 . 2  |-  ( E. x  e.  A  C  C_  B  ->  ( y  e.  C  ->  y  e. 
U_ x  e.  A  B ) )
76ssrdv 3176 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 2160   E.wrex 2469    C_ wss 3144   U_ciun 3901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-iun 3903
This theorem is referenced by:  iunss2  3946  iunpwss  3993  iunpw  4498
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