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Theorem difin0ss 3897
Description: Difference, intersection, and subclass relationship. (Contributed by NM, 30-Apr-1994.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
Assertion
Ref Expression
difin0ss  |-  ( ( ( A  \  B
)  i^i  C )  =  (/)  ->  ( C  C_  A  ->  C  C_  B
) )

Proof of Theorem difin0ss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eq0 3809 . 2  |-  ( ( ( A  \  B
)  i^i  C )  =  (/)  <->  A. x  -.  x  e.  ( ( A  \  B )  i^i  C
) )
2 iman 424 . . . . . 6  |-  ( ( x  e.  C  -> 
( x  e.  A  ->  x  e.  B ) )  <->  -.  ( x  e.  C  /\  -.  (
x  e.  A  ->  x  e.  B )
) )
3 elin 3683 . . . . . . . 8  |-  ( x  e.  ( ( A 
\  B )  i^i 
C )  <->  ( x  e.  ( A  \  B
)  /\  x  e.  C ) )
4 eldif 3481 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
54anbi1i 695 . . . . . . . 8  |-  ( ( x  e.  ( A 
\  B )  /\  x  e.  C )  <->  ( ( x  e.  A  /\  -.  x  e.  B
)  /\  x  e.  C ) )
63, 5bitri 249 . . . . . . 7  |-  ( x  e.  ( ( A 
\  B )  i^i 
C )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  /\  x  e.  C ) )
7 ancom 450 . . . . . . 7  |-  ( ( x  e.  C  /\  ( x  e.  A  /\  -.  x  e.  B
) )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  /\  x  e.  C ) )
8 annim 425 . . . . . . . 8  |-  ( ( x  e.  A  /\  -.  x  e.  B
)  <->  -.  ( x  e.  A  ->  x  e.  B ) )
98anbi2i 694 . . . . . . 7  |-  ( ( x  e.  C  /\  ( x  e.  A  /\  -.  x  e.  B
) )  <->  ( x  e.  C  /\  -.  (
x  e.  A  ->  x  e.  B )
) )
106, 7, 93bitr2i 273 . . . . . 6  |-  ( x  e.  ( ( A 
\  B )  i^i 
C )  <->  ( x  e.  C  /\  -.  (
x  e.  A  ->  x  e.  B )
) )
112, 10xchbinxr 311 . . . . 5  |-  ( ( x  e.  C  -> 
( x  e.  A  ->  x  e.  B ) )  <->  -.  x  e.  ( ( A  \  B )  i^i  C
) )
12 ax-2 7 . . . . 5  |-  ( ( x  e.  C  -> 
( x  e.  A  ->  x  e.  B ) )  ->  ( (
x  e.  C  ->  x  e.  A )  ->  ( x  e.  C  ->  x  e.  B ) ) )
1311, 12sylbir 213 . . . 4  |-  ( -.  x  e.  ( ( A  \  B )  i^i  C )  -> 
( ( x  e.  C  ->  x  e.  A )  ->  (
x  e.  C  ->  x  e.  B )
) )
1413al2imi 1637 . . 3  |-  ( A. x  -.  x  e.  ( ( A  \  B
)  i^i  C )  ->  ( A. x ( x  e.  C  ->  x  e.  A )  ->  A. x ( x  e.  C  ->  x  e.  B ) ) )
15 dfss2 3488 . . 3  |-  ( C 
C_  A  <->  A. x
( x  e.  C  ->  x  e.  A ) )
16 dfss2 3488 . . 3  |-  ( C 
C_  B  <->  A. x
( x  e.  C  ->  x  e.  B ) )
1714, 15, 163imtr4g 270 . 2  |-  ( A. x  -.  x  e.  ( ( A  \  B
)  i^i  C )  ->  ( C  C_  A  ->  C  C_  B )
)
181, 17sylbi 195 1  |-  ( ( ( A  \  B
)  i^i  C )  =  (/)  ->  ( C  C_  A  ->  C  C_  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369   A.wal 1393    = wceq 1395    e. wcel 1819    \ cdif 3468    i^i cin 3470    C_ wss 3471   (/)c0 3793
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-or 370  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-ne 2654  df-v 3111  df-dif 3474  df-in 3478  df-ss 3485  df-nul 3794
This theorem is referenced by:  tz7.7  4913  tfi  6687  lebnumlem3  21589
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