Theorem intss1 3859

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 2740	. . . 4 |- x e. _V
2	1	elint 3850	. . 3 |- ( x e. |^| B <-> A. y ( y e. B -> x e. y ) )
3		eleq1 2240	. . . . . 6 |- ( y = A -> ( y e. B <-> A e. B ) )
4		eleq2 2241	. . . . . 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 2824	. . . 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 3161	1 |- ( A e. B -> |^| B C_ A )

Syntax hints:   -> wi 4   A.wal 1351   = wceq 1353   e. wcel 2148   C_ wss 3129   |^|cint 3844

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-v 2739  df-in 3135  df-ss 3142  df-int 3845

This theorem is referenced by:  intminss  3869  intmin3  3871  intab  3873  int0el  3874  trintssm  4116  inteximm  4148  onnmin  4566  peano5  4596  peano5nnnn  7888  peano5nni  8918  dfuzi  9359  bj-intabssel  14401  bj-intabssel1  14402