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Theorem ssiun 4313
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 3451 . . . . 5  |-  ( C 
C_  B  ->  (
y  e.  C  -> 
y  e.  B ) )
21reximi 2922 . . . 4  |-  ( E. x  e.  A  C  C_  B  ->  E. x  e.  A  ( y  e.  C  ->  y  e.  B ) )
3 r19.37av 2969 . . . 4  |-  ( E. x  e.  A  ( y  e.  C  -> 
y  e.  B )  ->  ( y  e.  C  ->  E. x  e.  A  y  e.  B ) )
42, 3syl 16 . . 3  |-  ( E. x  e.  A  C  C_  B  ->  ( y  e.  C  ->  E. x  e.  A  y  e.  B ) )
5 eliun 4276 . . 3  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
64, 5syl6ibr 227 . 2  |-  ( E. x  e.  A  C  C_  B  ->  ( y  e.  C  ->  y  e. 
U_ x  e.  A  B ) )
76ssrdv 3463 1  |-  ( E. x  e.  A  C  C_  B  ->  C  C_  U_ x  e.  A  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758   E.wrex 2796    C_ wss 3429   U_ciun 4272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-v 3073  df-in 3436  df-ss 3443  df-iun 4274
This theorem is referenced by:  iunss2  4316  iunpwss  4361  iunpw  6493  onfununi  6905  oen0  7128  trcl  8052  rtrclreclem.refl  27483  rtrclreclem.subset  27484  trpredtr  27831  dftrpred3g  27834  wfrlem9  27869  frrlem5e  27913
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