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.

Step	Hyp	Ref	Expression
1		vex 2774	. . . 4 |- x e. _V
2	1	elint 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 ) )
5	3, 4	imbi12d 234	. . . . 5 |- ( y = A -> ( ( y e. B -> x e. y ) <-> ( A e. B -> x e. A ) ) )
6	5	spcgv 2859	. . . 4 |- ( A e. B -> ( A. y ( y e. B -> x e. y ) -> ( A e. B -> x e. A ) ) )
7	6	pm2.43a 51	. . 3 |- ( A e. B -> ( A. y ( y e. B -> x e. y ) -> x e. A ) )
8	2, 7	biimtrid 152	. 2 |- ( A e. B -> ( x e. |^| B -> x e. A ) )
9	8	ssrdv 3198	1 |- ( A e. B -> |^| B C_ A )

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