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Theorem ssiun2 4324
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 2847 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  E. x  e.  A  y  e.  B )
21ex 436 . . 3  |-  ( x  e.  A  ->  (
y  e.  B  ->  E. x  e.  A  y  e.  B )
)
3 eliun 4286 . . 3  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
42, 3syl6ibr 231 . 2  |-  ( x  e.  A  ->  (
y  e.  B  -> 
y  e.  U_ x  e.  A  B )
)
54ssrdv 3440 1  |-  ( x  e.  A  ->  B  C_ 
U_ x  e.  A  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1889   E.wrex 2740    C_ wss 3406   U_ciun 4281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ral 2744  df-rex 2745  df-v 3049  df-in 3413  df-ss 3420  df-iun 4283
This theorem is referenced by:  ssiun2s  4325  disjxiun  4402  triun  4513  ixpf  7549  ixpiunwdom  8111  r1sdom  8250  r1val1  8262  rankuni2b  8329  rankval4  8343  cplem1  8365  domtriomlem  8877  ac6num  8914  iunfo  8969  iundom2g  8970  pwfseqlem3  9090  inar1  9205  tskuni  9213  iunconlem  20454  ptclsg  20642  ovoliunlem1  22467  limciun  22861  ssiun2sf  28186  bnj906  29753  bnj999  29780  bnj1014  29783  bnj1408  29857  trpredrec  30491  sge0iunmpt  38270  sge0iun  38271  omeiunltfirp  38350  iunopeqop  39015
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