MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssiun2 Structured version   Unicode version

Theorem ssiun2 4375
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 2915 . . . 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 4337 . . 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 3505 1  |-  ( x  e.  A  ->  B  C_ 
U_ x  e.  A  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1819   E.wrex 2808    C_ wss 3471   U_ciun 4332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-in 3478  df-ss 3485  df-iun 4334
This theorem is referenced by:  ssiun2s  4376  disjxiun  4453  triun  4563  ixpf  7510  ixpiunwdom  8035  r1sdom  8209  r1val1  8221  rankuni2b  8288  rankval4  8302  cplem1  8324  domtriomlem  8839  ac6num  8876  iunfo  8931  iundom2g  8932  pwfseqlem3  9055  inar1  9170  tskuni  9178  iunconlem  20054  ptclsg  20242  ovoliunlem1  22039  limciun  22424  ssiun2sf  27565  trpredrec  29538  bnj906  34131  bnj999  34158  bnj1014  34161  bnj1408  34235
  Copyright terms: Public domain W3C validator