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Theorem intss 4063
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 72 . . . . 5  |-  ( ( y  e.  A  -> 
y  e.  B )  ->  ( ( y  e.  B  ->  x  e.  y )  ->  (
y  e.  A  ->  x  e.  y )
) )
21al2imi 1570 . . . 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 2951 . . . . 5  |-  x  e. 
_V
43elint 4048 . . . 4  |-  ( x  e.  |^| B  <->  A. y
( y  e.  B  ->  x  e.  y ) )
53elint 4048 . . . 4  |-  ( x  e.  |^| A  <->  A. y
( y  e.  A  ->  x  e.  y ) )
62, 4, 53imtr4g 262 . . 3  |-  ( A. y ( y  e.  A  ->  y  e.  B )  ->  (
x  e.  |^| B  ->  x  e.  |^| A
) )
76alrimiv 1641 . 2  |-  ( A. y ( y  e.  A  ->  y  e.  B )  ->  A. x
( x  e.  |^| B  ->  x  e.  |^| A ) )
8 dfss2 3329 . 2  |-  ( A 
C_  B  <->  A. y
( y  e.  A  ->  y  e.  B ) )
9 dfss2 3329 . 2  |-  ( |^| B  C_  |^| A  <->  A. x
( x  e.  |^| B  ->  x  e.  |^| A ) )
107, 8, 93imtr4i 258 1  |-  ( A 
C_  B  ->  |^| B  C_ 
|^| A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549    e. wcel 1725    C_ wss 3312   |^|cint 4042
This theorem is referenced by:  uniintsn  4079  intabs  4353  fiss  7421  tc2  7673  tcss  7675  tcel  7676  rankval4  7785  cfub  8121  cflm  8122  cflecard  8125  fin23lem26  8197  mrcss  13833  lspss  16052  lbsextlem3  16224  aspss  16383  clsss  17110  1stcfb  17500  ufinffr  17953  spanss  22842  pclssN  30628  dochspss  32113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319  df-ss 3326  df-int 4043
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