Theorem intss1 3621

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 2554	. . . 4 |- x e. _V
2	1	elint 3612	. . 3 |- (x e. |^|B <-> A.y(y e. B -> x e. y))
3		eleq1 2097	. . . . . 6 |- (y = A -> (y e. B <-> A e. B))
4		eleq2 2098	. . . . . 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 2634	. . . 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 2945	1 |- (A e. B -> |^|B C_ A)

Syntax hints:   -> wi 4  A.wal 1240   = wceq 1242   e. wcel 1390   C_ wss 2911   |^|cint 3606

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-int 3607

This theorem is referenced by:  intminss  3631  intmin3  3633  intab  3635  int0el  3636  trint0m  3862  inteximm  3894  onnmin  4244  peano5  4264  peano5nni  7698  dfuzi  8124  bj-intabssel  9263  bj-intabssel1  9264  peano5setOLD  9400