ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ss2iun Unicode version

Theorem ss2iun 3931
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 3177 . . . . 5  |-  ( B 
C_  C  ->  (
y  e.  B  -> 
y  e.  C ) )
21ralimi 2560 . . . 4  |-  ( A. x  e.  A  B  C_  C  ->  A. x  e.  A  ( y  e.  B  ->  y  e.  C ) )
3 rexim 2591 . . . 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 14 . . 3  |-  ( A. x  e.  A  B  C_  C  ->  ( E. x  e.  A  y  e.  B  ->  E. x  e.  A  y  e.  C ) )
5 eliun 3920 . . 3  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
6 eliun 3920 . . 3  |-  ( y  e.  U_ x  e.  A  C  <->  E. x  e.  A  y  e.  C )
74, 5, 63imtr4g 205 . 2  |-  ( A. x  e.  A  B  C_  C  ->  ( y  e.  U_ x  e.  A  B  ->  y  e.  U_ x  e.  A  C
) )
87ssrdv 3189 1  |-  ( A. x  e.  A  B  C_  C  ->  U_ x  e.  A  B  C_  U_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2167   A.wral 2475   E.wrex 2476    C_ wss 3157   U_ciun 3916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-in 3163  df-ss 3170  df-iun 3918
This theorem is referenced by:  iuneq2  3932  abnexg  4481  imasaddvallemg  12958  dvfvalap  14917
  Copyright terms: Public domain W3C validator