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Theorem ssiun2 3914
Description: Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ssiun2  |-  ( x  e.  A  ->  B  C_ 
U_ x  e.  A  B )

Proof of Theorem ssiun2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 rspe 2519 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  E. x  e.  A  y  e.  B )
21ex 114 . . 3  |-  ( x  e.  A  ->  (
y  e.  B  ->  E. x  e.  A  y  e.  B )
)
3 eliun 3875 . . 3  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
42, 3syl6ibr 161 . 2  |-  ( x  e.  A  ->  (
y  e.  B  -> 
y  e.  U_ x  e.  A  B )
)
54ssrdv 3153 1  |-  ( x  e.  A  ->  B  C_ 
U_ x  e.  A  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2141   E.wrex 2449    C_ wss 3121   U_ciun 3871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-iun 3873
This theorem is referenced by:  ssiun2s  3915  triun  4098  ixpf  6695
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