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Theorem iinuniss 3969
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 2625 . . . 4  |-  ( ( y  e.  A  \/  A. x  e.  B  y  e.  x )  ->  A. x  e.  B  ( y  e.  A  \/  y  e.  x
) )
2 vex 2740 . . . . . 6  |-  y  e. 
_V
32elint2 3851 . . . . 5  |-  ( y  e.  |^| B  <->  A. x  e.  B  y  e.  x )
43orbi2i 762 . . . 4  |-  ( ( y  e.  A  \/  y  e.  |^| B )  <-> 
( y  e.  A  \/  A. x  e.  B  y  e.  x )
)
5 elun 3276 . . . . 5  |-  ( y  e.  ( A  u.  x )  <->  ( y  e.  A  \/  y  e.  x ) )
65ralbii 2483 . . . 4  |-  ( A. x  e.  B  y  e.  ( A  u.  x
)  <->  A. x  e.  B  ( y  e.  A  \/  y  e.  x
) )
71, 4, 63imtr4i 201 . . 3  |-  ( ( y  e.  A  \/  y  e.  |^| B )  ->  A. x  e.  B  y  e.  ( A  u.  x ) )
87ss2abi 3227 . 2  |-  { y  |  ( y  e.  A  \/  y  e. 
|^| B ) } 
C_  { y  | 
A. x  e.  B  y  e.  ( A  u.  x ) }
9 df-un 3133 . 2  |-  ( A  u.  |^| B )  =  { y  |  ( y  e.  A  \/  y  e.  |^| B ) }
10 df-iin 3889 . 2  |-  |^|_ x  e.  B  ( A  u.  x )  =  {
y  |  A. x  e.  B  y  e.  ( A  u.  x
) }
118, 9, 103sstr4i 3196 1  |-  ( A  u.  |^| B )  C_  |^|_
x  e.  B  ( A  u.  x )
Colors of variables: wff set class
Syntax hints:    \/ wo 708    e. wcel 2148   {cab 2163   A.wral 2455    u. cun 3127    C_ wss 3129   |^|cint 3844   |^|_ciin 3887
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-int 3845  df-iin 3889
This theorem is referenced by: (None)
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