Theorem intss1 3964

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 2816	. . . 4 |- x e. _V
2	1	elint 3955	. . 3 |- ( x e. |^| B <-> A. y ( y e. B -> x e. y ) )
3		eleq1 2295	. . . . . 6 |- ( y = A -> ( y e. B <-> A e. B ) )
4		eleq2 2296	. . . . . 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 2904	. . . 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 3244	1 |- ( A e. B -> |^| B C_ A )

Syntax hints:   -> wi 4   A.wal 1396   = wceq 1398   e. wcel 2203   C_ wss 3211   |^|cint 3949

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217  df-ss 3224  df-int 3950

This theorem is referenced by:  intminss  3974  intmin3  3976  intab  3978  int0el  3979  trintssm  4224  inteximm  4261  onnmin  4690  peano5  4720  peano5nnnn  8207  peano5nni  9240  dfuzi  9688  bj-intabssel  16561  bj-intabssel1  16562