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Theorem iinuniss 3811
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 2515 . . . 4  |-  ( ( y  e.  A  \/  A. x  e.  B  y  e.  x )  ->  A. x  e.  B  ( y  e.  A  \/  y  e.  x
) )
2 vex 2622 . . . . . 6  |-  y  e. 
_V
32elint2 3695 . . . . 5  |-  ( y  e.  |^| B  <->  A. x  e.  B  y  e.  x )
43orbi2i 714 . . . 4  |-  ( ( y  e.  A  \/  y  e.  |^| B )  <-> 
( y  e.  A  \/  A. x  e.  B  y  e.  x )
)
5 elun 3141 . . . . 5  |-  ( y  e.  ( A  u.  x )  <->  ( y  e.  A  \/  y  e.  x ) )
65ralbii 2384 . . . 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 3093 . 2  |-  { y  |  ( y  e.  A  \/  y  e. 
|^| B ) } 
C_  { y  | 
A. x  e.  B  y  e.  ( A  u.  x ) }
9 df-un 3003 . 2  |-  ( A  u.  |^| B )  =  { y  |  ( y  e.  A  \/  y  e.  |^| B ) }
10 df-iin 3733 . 2  |-  |^|_ x  e.  B  ( A  u.  x )  =  {
y  |  A. x  e.  B  y  e.  ( A  u.  x
) }
118, 9, 103sstr4i 3065 1  |-  ( A  u.  |^| B )  C_  |^|_
x  e.  B  ( A  u.  x )
Colors of variables: wff set class
Syntax hints:    \/ wo 664    e. wcel 1438   {cab 2074   A.wral 2359    u. cun 2997    C_ wss 2999   |^|cint 3688   |^|_ciin 3731
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-int 3689  df-iin 3733
This theorem is referenced by: (None)
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