Theorem intss 3877

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 1468	. . . 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 2752	. . . . 5 |- x e. _V
4	3	elint 3862	. . . 4 |- ( x e. |^| B <-> A. y ( y e. B -> x e. y ) )
5	3	elint 3862	. . . 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 1884	. 2 |- ( A. y ( y e. A -> y e. B ) -> A. x ( x e. |^| B -> x e. |^| A ) )
8		dfss2 3156	. 2 |- ( A C_ B <-> A. y ( y e. A -> y e. B ) )
9		dfss2 3156	. 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 1361   e. wcel 2158   C_ wss 3141   |^|cint 3856

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-in 3147  df-ss 3154  df-int 3857

This theorem is referenced by:  lspss  13583  clsss  13889