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

Theorem iinuniss 3766
Description: A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33 but with equality changed to subset. (Contributed by Jim Kingdon, 19-Aug-2018.)
Assertion
Ref Expression
iinuniss  |-  ( A  u.  |^| B )  C_  |^|_
x  e.  B  ( A  u.  x )
Distinct variable groups:    x, A    x, B

Proof of Theorem iinuniss
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 r19.32vr 2503 . . . 4  |-  ( ( y  e.  A  \/  A. x  e.  B  y  e.  x )  ->  A. x  e.  B  ( y  e.  A  \/  y  e.  x
) )
2 vex 2605 . . . . . 6  |-  y  e. 
_V
32elint2 3651 . . . . 5  |-  ( y  e.  |^| B  <->  A. x  e.  B  y  e.  x )
43orbi2i 712 . . . 4  |-  ( ( y  e.  A  \/  y  e.  |^| B )  <-> 
( y  e.  A  \/  A. x  e.  B  y  e.  x )
)
5 elun 3114 . . . . 5  |-  ( y  e.  ( A  u.  x )  <->  ( y  e.  A  \/  y  e.  x ) )
65ralbii 2373 . . . 4  |-  ( A. x  e.  B  y  e.  ( A  u.  x
)  <->  A. x  e.  B  ( y  e.  A  \/  y  e.  x
) )
71, 4, 63imtr4i 199 . . 3  |-  ( ( y  e.  A  \/  y  e.  |^| B )  ->  A. x  e.  B  y  e.  ( A  u.  x ) )
87ss2abi 3067 . 2  |-  { y  |  ( y  e.  A  \/  y  e. 
|^| B ) } 
C_  { y  | 
A. x  e.  B  y  e.  ( A  u.  x ) }
9 df-un 2978 . 2  |-  ( A  u.  |^| B )  =  { y  |  ( y  e.  A  \/  y  e.  |^| B ) }
10 df-iin 3689 . 2  |-  |^|_ x  e.  B  ( A  u.  x )  =  {
y  |  A. x  e.  B  y  e.  ( A  u.  x
) }
118, 9, 103sstr4i 3039 1  |-  ( A  u.  |^| B )  C_  |^|_
x  e.  B  ( A  u.  x )
Colors of variables: wff set class
Syntax hints:    \/ wo 662    e. wcel 1434   {cab 2068   A.wral 2349    u. cun 2972    C_ wss 2974   |^|cint 3644   |^|_ciin 3687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-int 3645  df-iin 3689
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator