Theorem intss 3891

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 1469	. . . 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 2763	. . . . 5 |- x e. _V
4	3	elint 3876	. . . 4 |- ( x e. |^| B <-> A. y ( y e. B -> x e. y ) )
5	3	elint 3876	. . . 4 |- ( x e. |^| A <-> A. y ( y e. A -> x e. y ) )
6	2, 4, 5	3imtr4g 205	. . 3 |- ( A. y ( y e. A -> y e. B ) -> ( x e. |^| B -> x e. |^| A ) )
7	6	alrimiv 1885	. 2 |- ( A. y ( y e. A -> y e. B ) -> A. x ( x e. |^| B -> x e. |^| A ) )
8		dfss2 3168	. 2 |- ( A C_ B <-> A. y ( y e. A -> y e. B ) )
9		dfss2 3168	. 2 |- ( |^| B C_ |^| A <-> A. x ( x e. |^| B -> x e. |^| A ) )
10	7, 8, 9	3imtr4i 201	1 |- ( A C_ B -> |^| B C_ |^| A )

Syntax hints:   -> wi 4   A.wal 1362   e. wcel 2164   C_ wss 3153   |^|cint 3870

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159  df-ss 3166  df-int 3871

This theorem is referenced by:  lspss  13895  clsss  14286