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Theorem intssOLD 4310
Description: Intersection of subclasses. (Contributed by NM, 14-Oct-1999.) Obsolete version of intss 4309 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
intssOLD  |-  ( A 
C_  B  ->  |^| B  C_ 
|^| A )

Proof of Theorem intssOLD
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 1637 . . . 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 3112 . . . . 5  |-  x  e. 
_V
43elint 4294 . . . 4  |-  ( x  e.  |^| B  <->  A. y
( y  e.  B  ->  x  e.  y ) )
53elint 4294 . . . 4  |-  ( x  e.  |^| A  <->  A. y
( y  e.  A  ->  x  e.  y ) )
62, 4, 53imtr4g 270 . . 3  |-  ( A. y ( y  e.  A  ->  y  e.  B )  ->  (
x  e.  |^| B  ->  x  e.  |^| A
) )
76alrimiv 1720 . 2  |-  ( A. y ( y  e.  A  ->  y  e.  B )  ->  A. x
( x  e.  |^| B  ->  x  e.  |^| A ) )
8 dfss2 3488 . 2  |-  ( A 
C_  B  <->  A. y
( y  e.  A  ->  y  e.  B ) )
9 dfss2 3488 . 2  |-  ( |^| B  C_  |^| A  <->  A. x
( x  e.  |^| B  ->  x  e.  |^| A ) )
107, 8, 93imtr4i 266 1  |-  ( A 
C_  B  ->  |^| B  C_ 
|^| A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1393    e. wcel 1819    C_ wss 3471   |^|cint 4288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-in 3478  df-ss 3485  df-int 4289
This theorem is referenced by: (None)
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