Theorem ssint 3900

Description: Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)

Assertion

Ref	Expression
ssint	|- ( A C_ |^| B <-> A. x e. B A C_ x )

Distinct variable groups:   x, A   x, B

Proof of Theorem ssint

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss3 3181	. 2 |- ( A C_ |^| B <-> A. y e. A y e. |^| B )
2		vex 2774	. . . 4 |- y e. _V
3	2	elint2 3891	. . 3 |- ( y e. |^| B <-> A. x e. B y e. x )
4	3	ralbii 2511	. 2 |- ( A. y e. A y e. |^| B <-> A. y e. A A. x e. B y e. x )
5		ralcom 2668	. . 3 |- ( A. y e. A A. x e. B y e. x <-> A. x e. B A. y e. A y e. x )
6		dfss3 3181	. . . 4 |- ( A C_ x <-> A. y e. A y e. x )
7	6	ralbii 2511	. . 3 |- ( A. x e. B A C_ x <-> A. x e. B A. y e. A y e. x )
8	5, 7	bitr4i 187	. 2 |- ( A. y e. A A. x e. B y e. x <-> A. x e. B A C_ x )
9	1, 4, 8	3bitri 206	1 |- ( A C_ |^| B <-> A. x e. B A C_ x )

Syntax hints:   <-> wb 105   e. wcel 2175   A.wral 2483   C_ wss 3165   |^|cint 3884

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-v 2773  df-in 3171  df-ss 3178  df-int 3885

This theorem is referenced by:  ssintab  3901  ssintub  3902  iinpw  4017  trint  4156  fintm  5460  bj-ssom  15805