Theorem intss1 3941

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 2862	. . . 4 |- x e. _V
2	1	elint 3932	. . 3 |- (x e. |^|B <-> A.y(y e. B -> x e. y))
3		eleq1 2413	. . . . . 6 |- (y = A -> (y e. B <-> A e. B))
4		eleq2 2414	. . . . . 6 |- (y = A -> (x e. y <-> x e. A))
5	3, 4	imbi12d 311	. . . . 5 |- (y = A -> ((y e. B -> x e. y) <-> (A e. B -> x e. A)))
6	5	spcgv 2939	. . . 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 208	. 2 |- (A e. B -> (x e. |^|B -> x e. A))
9	8	ssrdv 3278	1 |- (A e. B -> |^|B C_ A)

Syntax hints:   -> wi 4  A.wal 1540   = wceq 1642   e. wcel 1710   C_ wss 3257  |^|cint 3926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-int 3927

This theorem is referenced by:  intminss  3952  intmin3  3954  intab  3956  int0el  3957  peano5  4409  spfininduct  4540  clos1induct  5880