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Theorem ssiun 3944
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 3174 . . . . 5  |-  ( C 
C_  B  ->  (
y  e.  C  -> 
y  e.  B ) )
21reximi 2650 . . . 4  |-  ( E. x  e.  A  C  C_  B  ->  E. x  e.  A  ( y  e.  C  ->  y  e.  B ) )
3 r19.37av 2690 . . . 4  |-  ( E. x  e.  A  ( y  e.  C  -> 
y  e.  B )  ->  ( y  e.  C  ->  E. x  e.  A  y  e.  B ) )
42, 3syl 15 . . 3  |-  ( E. x  e.  A  C  C_  B  ->  ( y  e.  C  ->  E. x  e.  A  y  e.  B ) )
5 eliun 3909 . . 3  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
64, 5syl6ibr 218 . 2  |-  ( E. x  e.  A  C  C_  B  ->  ( y  e.  C  ->  y  e. 
U_ x  e.  A  B ) )
76ssrdv 3185 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 1684   E.wrex 2544    C_ wss 3152   U_ciun 3905
This theorem is referenced by:  iunss2  3947  iunpwss  3991  iunpw  4570  onfununi  6358  oen0  6584  trcl  7410  rtrclreclem.refl  23452  rtrclreclem.subset  23453  trpredtr  23644  dftrpred3g  23647  wfrlem9  23675  frrlem5e  23700
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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