MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ss2iun Structured version   Unicode version

Theorem ss2iun 4312
Description: Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ss2iun  |-  ( A. x  e.  A  B  C_  C  ->  U_ x  e.  A  B  C_  U_ x  e.  A  C )

Proof of Theorem ss2iun
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssel 3458 . . . . 5  |-  ( B 
C_  C  ->  (
y  e.  B  -> 
y  e.  C ) )
21ralimi 2818 . . . 4  |-  ( A. x  e.  A  B  C_  C  ->  A. x  e.  A  ( y  e.  B  ->  y  e.  C ) )
3 rexim 2890 . . . 4  |-  ( A. x  e.  A  (
y  e.  B  -> 
y  e.  C )  ->  ( E. x  e.  A  y  e.  B  ->  E. x  e.  A  y  e.  C )
)
42, 3syl 17 . . 3  |-  ( A. x  e.  A  B  C_  C  ->  ( E. x  e.  A  y  e.  B  ->  E. x  e.  A  y  e.  C ) )
5 eliun 4301 . . 3  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
6 eliun 4301 . . 3  |-  ( y  e.  U_ x  e.  A  C  <->  E. x  e.  A  y  e.  C )
74, 5, 63imtr4g 273 . 2  |-  ( A. x  e.  A  B  C_  C  ->  ( y  e.  U_ x  e.  A  B  ->  y  e.  U_ x  e.  A  C
) )
87ssrdv 3470 1  |-  ( A. x  e.  A  B  C_  C  ->  U_ x  e.  A  B  C_  U_ x  e.  A  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1868   A.wral 2775   E.wrex 2776    C_ wss 3436   U_ciun 4296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ral 2780  df-rex 2781  df-v 3083  df-in 3443  df-ss 3450  df-iun 4298
This theorem is referenced by:  iuneq2  4313  oawordri  7256  omwordri  7278  oewordri  7298  oeworde  7299  r1val1  8259  cfslb2n  8699  imasaddvallem  15423  dprdss  17650  tgcmp  20403  txcmplem1  20643  txcmplem2  20644  xkococnlem  20661  alexsubALT  21053  ptcmplem3  21056  metnrmlem2  21864  metnrmlem2OLD  21879  uniiccvol  22524  dvfval  22839  bnj1145  29798  bnj1136  29802  filnetlem3  31029  poimirlem32  31886  sstotbnd2  32020  equivtotbnd  32024  trclrelexplem  36163  corcltrcl  36191  cotrclrcl  36194
  Copyright terms: Public domain W3C validator