Theorem intss1 3914

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 2779	. . . 4 |- x e. _V
2	1	elint 3905	. . 3 |- ( x e. |^| B <-> A. y ( y e. B -> x e. y ) )
3		eleq1 2270	. . . . . 6 |- ( y = A -> ( y e. B <-> A e. B ) )
4		eleq2 2271	. . . . . 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 2867	. . . 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 3207	1 |- ( A e. B -> |^| B C_ A )

Syntax hints:   -> wi 4   A.wal 1371   = wceq 1373   e. wcel 2178   C_ wss 3174   |^|cint 3899

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-ss 3187  df-int 3900

This theorem is referenced by:  intminss  3924  intmin3  3926  intab  3928  int0el  3929  trintssm  4174  inteximm  4209  onnmin  4634  peano5  4664  peano5nnnn  8040  peano5nni  9074  dfuzi  9518  bj-intabssel  15925  bj-intabssel1  15926