Theorem intss1 3985

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 2824	. . . 4 |- x e. _V
2	1	elint 3976	. . 3 |- ( x e. |^| B <-> A. y ( y e. B -> x e. y ) )
3		eleq1 2301	. . . . . 6 |- ( y = A -> ( y e. B <-> A e. B ) )
4		eleq2 2302	. . . . . 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 2912	. . . 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 3254	1 |- ( A e. B -> |^| B C_ A )

This proof depends on syntax axioms:   -> wi 4   A.wal 1400   = wceq 1402   e. wcel 2209   C_ wss 3220   |^|cint 3970

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-int 3971

This theorem is used by:  intminss  3995  intmin3  3997  intab  3999  int0el  4000  trintssm  4245  inteximm  4285  onnmin  4715  peano5  4745  peano5nnnn  8259  peano5nni  9307  dfuzi  9756  bj-intabssel  16817  bj-intabssel1  16818