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Theorem intss 3899
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 70 . . . . 5  |-  ( ( y  e.  A  -> 
y  e.  B )  ->  ( ( y  e.  B  ->  x  e.  y )  ->  (
y  e.  A  ->  x  e.  y )
) )
21al2imi 1551 . . . 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 2804 . . . . 5  |-  x  e. 
_V
43elint 3884 . . . 4  |-  ( x  e.  |^| B  <->  A. y
( y  e.  B  ->  x  e.  y ) )
53elint 3884 . . . 4  |-  ( x  e.  |^| A  <->  A. y
( y  e.  A  ->  x  e.  y ) )
62, 4, 53imtr4g 261 . . 3  |-  ( A. y ( y  e.  A  ->  y  e.  B )  ->  (
x  e.  |^| B  ->  x  e.  |^| A
) )
76alrimiv 1621 . 2  |-  ( A. y ( y  e.  A  ->  y  e.  B )  ->  A. x
( x  e.  |^| B  ->  x  e.  |^| A ) )
8 dfss2 3182 . 2  |-  ( A 
C_  B  <->  A. y
( y  e.  A  ->  y  e.  B ) )
9 dfss2 3182 . 2  |-  ( |^| B  C_  |^| A  <->  A. x
( x  e.  |^| B  ->  x  e.  |^| A ) )
107, 8, 93imtr4i 257 1  |-  ( A 
C_  B  ->  |^| B  C_ 
|^| A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530    e. wcel 1696    C_ wss 3165   |^|cint 3878
This theorem is referenced by:  uniintsn  3915  intabs  4188  fiss  7193  tc2  7443  tcss  7445  tcel  7446  rankval4  7555  cfub  7891  cflm  7892  cflecard  7895  fin23lem26  7967  mrcss  13534  lspss  15757  lbsextlem3  15929  aspss  16088  clsss  16807  1stcfb  17187  ufinffr  17640  spanss  21943  pclssN  30705  dochspss  32190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-ss 3179  df-int 3879
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