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Theorem ssiun2 4368
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 2922 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  E. x  e.  A  y  e.  B )
21ex 434 . . 3  |-  ( x  e.  A  ->  (
y  e.  B  ->  E. x  e.  A  y  e.  B )
)
3 eliun 4330 . . 3  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
42, 3syl6ibr 227 . 2  |-  ( x  e.  A  ->  (
y  e.  B  -> 
y  e.  U_ x  e.  A  B )
)
54ssrdv 3510 1  |-  ( x  e.  A  ->  B  C_ 
U_ x  e.  A  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   E.wrex 2815    C_ wss 3476   U_ciun 4325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-v 3115  df-in 3483  df-ss 3490  df-iun 4327
This theorem is referenced by:  ssiun2s  4369  disjxiun  4444  triun  4553  ixpf  7488  ixpiunwdom  8013  r1sdom  8188  r1val1  8200  rankuni2b  8267  rankval4  8281  cplem1  8303  domtriomlem  8818  ac6num  8855  iunfo  8910  iundom2g  8911  pwfseqlem3  9034  inar1  9149  tskuni  9157  iunconlem  19694  ptclsg  19851  ovoliunlem1  21648  limciun  22033  ssiun2sf  27100  trpredrec  28898  bnj906  33067  bnj999  33094  bnj1014  33097  bnj1408  33171
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