MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iunss1 Unicode version

Theorem iunss1 3932
Description: Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iunss1  |-  ( A 
C_  B  ->  U_ x  e.  A  C  C_  U_ x  e.  B  C )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem iunss1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssrexv 3251 . . 3  |-  ( A 
C_  B  ->  ( E. x  e.  A  y  e.  C  ->  E. x  e.  B  y  e.  C ) )
2 eliun 3925 . . 3  |-  ( y  e.  U_ x  e.  A  C  <->  E. x  e.  A  y  e.  C )
3 eliun 3925 . . 3  |-  ( y  e.  U_ x  e.  B  C  <->  E. x  e.  B  y  e.  C )
41, 2, 33imtr4g 261 . 2  |-  ( A 
C_  B  ->  (
y  e.  U_ x  e.  A  C  ->  y  e.  U_ x  e.  B  C ) )
54ssrdv 3198 1  |-  ( A 
C_  B  ->  U_ x  e.  A  C  C_  U_ x  e.  B  C )
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:  iuneq1  3934  iunxdif2  3966  oelim2  6609  fsumiun  12295  ovolfiniun  18876  uniioovol  18950  bnj1413  29381  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
  Copyright terms: Public domain W3C validator