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Theorem intss1 3899
Description: An element of a class includes the intersection of the class. Exercise 4 of [TakeutiZaring] p. 44 (with correction), generalized to classes. (Contributed by NM, 18-Nov-1995.)
Assertion
Ref Expression
intss1  |-  ( A  e.  B  ->  |^| B  C_  A )

Proof of Theorem intss1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2774 . . . 4  |-  x  e. 
_V
21elint 3890 . . 3  |-  ( x  e.  |^| B  <->  A. y
( y  e.  B  ->  x  e.  y ) )
3 eleq1 2267 . . . . . 6  |-  ( y  =  A  ->  (
y  e.  B  <->  A  e.  B ) )
4 eleq2 2268 . . . . . 6  |-  ( y  =  A  ->  (
x  e.  y  <->  x  e.  A ) )
53, 4imbi12d 234 . . . . 5  |-  ( y  =  A  ->  (
( y  e.  B  ->  x  e.  y )  <-> 
( A  e.  B  ->  x  e.  A ) ) )
65spcgv 2859 . . . 4  |-  ( A  e.  B  ->  ( A. y ( y  e.  B  ->  x  e.  y )  ->  ( A  e.  B  ->  x  e.  A ) ) )
76pm2.43a 51 . . 3  |-  ( A  e.  B  ->  ( A. y ( y  e.  B  ->  x  e.  y )  ->  x  e.  A ) )
82, 7biimtrid 152 . 2  |-  ( A  e.  B  ->  (
x  e.  |^| B  ->  x  e.  A ) )
98ssrdv 3198 1  |-  ( A  e.  B  ->  |^| B  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1370    = wceq 1372    e. wcel 2175    C_ wss 3165   |^|cint 3884
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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186
This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171  df-ss 3178  df-int 3885
This theorem is referenced by:  intminss  3909  intmin3  3911  intab  3913  int0el  3914  trintssm  4157  inteximm  4192  onnmin  4615  peano5  4645  peano5nnnn  8004  peano5nni  9038  dfuzi  9482  bj-intabssel  15687  bj-intabssel1  15688
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