Theorem intss1 3630

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 2560	. . . 4 |- x e. _V
2	1	elint 3621	. . 3 |- ( x e. |^| B <-> A. y ( y e. B -> x e. y ) )
3		eleq1 2100	. . . . . 6 |- ( y = A -> ( y e. B <-> A e. B ) )
4		eleq2 2101	. . . . . 6 |- ( y = A -> ( x e. y <-> x e. A ) )
5	3, 4	imbi12d 223	. . . . 5 |- ( y = A -> ( ( y e. B -> x e. y ) <-> ( A e. B -> x e. A ) ) )
6	5	spcgv 2640	. . . 4 |- ( A e. B -> ( A. y ( y e. B -> x e. y ) -> ( A e. B -> x e. A ) ) )
7	6	pm2.43a 45	. . 3 |- ( A e. B -> ( A. y ( y e. B -> x e. y ) -> x e. A ) )
8	2, 7	syl5bi 141	. 2 |- ( A e. B -> ( x e. |^| B -> x e. A ) )
9	8	ssrdv 2951	1 |- ( A e. B -> |^| B C_ A )

Syntax hints:   -> wi 4   A.wal 1241   = wceq 1243   e. wcel 1393   C_ wss 2917   |^|cint 3615

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-int 3616

This theorem is referenced by:  intminss  3640  intmin3  3642  intab  3644  int0el  3645  trint0m  3871  inteximm  3903  onnmin  4292  peano5  4321  peano5nnnn  6966  peano5nni  7917  dfuzi  8348  bj-intabssel  9928  bj-intabssel1  9929  peano5setOLD  10065