ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  intss Unicode version

Theorem intss 3852
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.
StepHypRef Expression
1 imim1 76 . . . . 5  |-  ( ( y  e.  A  -> 
y  e.  B )  ->  ( ( y  e.  B  ->  x  e.  y )  ->  (
y  e.  A  ->  x  e.  y )
) )
21al2imi 1451 . . . 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 2733 . . . . 5  |-  x  e. 
_V
43elint 3837 . . . 4  |-  ( x  e.  |^| B  <->  A. y
( y  e.  B  ->  x  e.  y ) )
53elint 3837 . . . 4  |-  ( x  e.  |^| A  <->  A. y
( y  e.  A  ->  x  e.  y ) )
62, 4, 53imtr4g 204 . . 3  |-  ( A. y ( y  e.  A  ->  y  e.  B )  ->  (
x  e.  |^| B  ->  x  e.  |^| A
) )
76alrimiv 1867 . 2  |-  ( A. y ( y  e.  A  ->  y  e.  B )  ->  A. x
( x  e.  |^| B  ->  x  e.  |^| A ) )
8 dfss2 3136 . 2  |-  ( A 
C_  B  <->  A. y
( y  e.  A  ->  y  e.  B ) )
9 dfss2 3136 . 2  |-  ( |^| B  C_  |^| A  <->  A. x
( x  e.  |^| B  ->  x  e.  |^| A ) )
107, 8, 93imtr4i 200 1  |-  ( A 
C_  B  ->  |^| B  C_ 
|^| A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1346    e. wcel 2141    C_ wss 3121   |^|cint 3831
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-int 3832
This theorem is referenced by:  clsss  12912
  Copyright terms: Public domain W3C validator