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Theorem ssiun2 3961
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 2617 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  E. x  e.  A  y  e.  B )
21ex 423 . . 3  |-  ( x  e.  A  ->  (
y  e.  B  ->  E. x  e.  A  y  e.  B )
)
3 eliun 3925 . . 3  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
42, 3syl6ibr 218 . 2  |-  ( x  e.  A  ->  (
y  e.  B  -> 
y  e.  U_ x  e.  A  B )
)
54ssrdv 3198 1  |-  ( x  e.  A  ->  B  C_ 
U_ x  e.  A  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   E.wrex 2557    C_ wss 3165   U_ciun 3921
This theorem is referenced by:  ssiun2s  3962  disjxiun  4036  triun  4142  ixpf  6854  ixpiunwdom  7321  r1sdom  7462  r1val1  7474  rankuni2b  7541  rankval4  7555  cplem1  7575  domtriomlem  8084  ac6num  8122  iunfo  8177  iundom2g  8178  pwfseqlem3  8298  inar1  8413  tskuni  8421  iunconlem  17169  ptclsg  17325  ovoliunlem1  18877  limciun  19260  ssiun2sf  23173  trpredrec  24312  bnj906  29278  bnj999  29305  bnj1014  29308  bnj1408  29382
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-in 3172  df-ss 3179  df-iun 3923
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