Theorem intss 3845

Description: Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)

Assertion

Ref	Expression
intss	|- ( A C_ B -> |^| B C_ |^| A )

Proof of Theorem intss

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imim1 76	. . . . 5 |- ( ( y e. A -> y e. B ) -> ( ( y e. B -> x e. y ) -> ( y e. A -> x e. y ) ) )
2	1	al2imi 1446	. . . 4 |- ( A. y ( y e. A -> y e. B ) -> ( A. y ( y e. B -> x e. y ) -> A. y ( y e. A -> x e. y ) ) )
3		vex 2729	. . . . 5 |- x e. _V
4	3	elint 3830	. . . 4 |- ( x e. |^| B <-> A. y ( y e. B -> x e. y ) )
5	3	elint 3830	. . . 4 |- ( x e. |^| A <-> A. y ( y e. A -> x e. y ) )
6	2, 4, 5	3imtr4g 204	. . 3 |- ( A. y ( y e. A -> y e. B ) -> ( x e. |^| B -> x e. |^| A ) )
7	6	alrimiv 1862	. 2 |- ( A. y ( y e. A -> y e. B ) -> A. x ( x e. |^| B -> x e. |^| A ) )
8		dfss2 3131	. 2 |- ( A C_ B <-> A. y ( y e. A -> y e. B ) )
9		dfss2 3131	. 2 |- ( |^| B C_ |^| A <-> A. x ( x e. |^| B -> x e. |^| A ) )
10	7, 8, 9	3imtr4i 200	1 |- ( A C_ B -> |^| B C_ |^| A )

Syntax hints:   -> wi 4   A.wal 1341   e. wcel 2136   C_ wss 3116   |^|cint 3824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-int 3825

This theorem is referenced by:  clsss  12758