Theorem intss 3948

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 70	. . . . 5 |- ((y e. A -> y e. B) -> ((y e. B -> x e. y) -> (y e. A -> x e. y)))
2	1	al2imi 1561	. . . 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 2863	. . . . 5 |- x e. _V
4	3	elint 3933	. . . 4 |- (x e. |^|B <-> A.y(y e. B -> x e. y))
5	3	elint 3933	. . . 4 |- (x e. |^|A <-> A.y(y e. A -> x e. y))
6	2, 4, 5	3imtr4g 261	. . 3 |- (A.y(y e. A -> y e. B) -> (x e. |^|B -> x e. |^|A))
7	6	alrimiv 1631	. 2 |- (A.y(y e. A -> y e. B) -> A.x(x e. |^|B -> x e. |^|A))
8		dfss2 3263	. 2 |- (A C_ B <-> A.y(y e. A -> y e. B))
9		dfss2 3263	. 2 |- (|^|B C_ |^|A <-> A.x(x e. |^|B -> x e. |^|A))
10	7, 8, 9	3imtr4i 257	1 |- (A C_ B -> |^|B C_ |^|A)

Syntax hints:   -> wi 4  A.wal 1540   e. wcel 1710   C_ wss 3258  |^|cint 3927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-int 3928

This theorem is referenced by:  uniintsn  3964