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Theorem difin0ss 3533
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 3482 . 2  |-  ( ( ( A  \  B
)  i^i  C )  =  (/)  <->  A. x  -.  x  e.  ( ( A  \  B )  i^i  C
) )
2 iman 413 . . . . . 6  |-  ( ( x  e.  C  -> 
( x  e.  A  ->  x  e.  B ) )  <->  -.  ( x  e.  C  /\  -.  (
x  e.  A  ->  x  e.  B )
) )
3 elin 3371 . . . . . . . 8  |-  ( x  e.  ( ( A 
\  B )  i^i 
C )  <->  ( x  e.  ( A  \  B
)  /\  x  e.  C ) )
4 eldif 3175 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
54anbi1i 676 . . . . . . . 8  |-  ( ( x  e.  ( A 
\  B )  /\  x  e.  C )  <->  ( ( x  e.  A  /\  -.  x  e.  B
)  /\  x  e.  C ) )
63, 5bitri 240 . . . . . . 7  |-  ( x  e.  ( ( A 
\  B )  i^i 
C )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  /\  x  e.  C ) )
7 ancom 437 . . . . . . 7  |-  ( ( x  e.  C  /\  ( x  e.  A  /\  -.  x  e.  B
) )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  /\  x  e.  C ) )
8 annim 414 . . . . . . . 8  |-  ( ( x  e.  A  /\  -.  x  e.  B
)  <->  -.  ( x  e.  A  ->  x  e.  B ) )
98anbi2i 675 . . . . . . 7  |-  ( ( x  e.  C  /\  ( x  e.  A  /\  -.  x  e.  B
) )  <->  ( x  e.  C  /\  -.  (
x  e.  A  ->  x  e.  B )
) )
106, 7, 93bitr2i 264 . . . . . 6  |-  ( x  e.  ( ( A 
\  B )  i^i 
C )  <->  ( x  e.  C  /\  -.  (
x  e.  A  ->  x  e.  B )
) )
112, 10xchbinxr 302 . . . . 5  |-  ( ( x  e.  C  -> 
( x  e.  A  ->  x  e.  B ) )  <->  -.  x  e.  ( ( A  \  B )  i^i  C
) )
12 ax-2 6 . . . . 5  |-  ( ( x  e.  C  -> 
( x  e.  A  ->  x  e.  B ) )  ->  ( (
x  e.  C  ->  x  e.  A )  ->  ( x  e.  C  ->  x  e.  B ) ) )
1311, 12sylbir 204 . . . 4  |-  ( -.  x  e.  ( ( A  \  B )  i^i  C )  -> 
( ( x  e.  C  ->  x  e.  A )  ->  (
x  e.  C  ->  x  e.  B )
) )
1413al2imi 1551 . . 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 3182 . . 3  |-  ( C 
C_  A  <->  A. x
( x  e.  C  ->  x  e.  A ) )
16 dfss2 3182 . . 3  |-  ( C 
C_  B  <->  A. x
( x  e.  C  ->  x  e.  B ) )
1714, 15, 163imtr4g 261 . 2  |-  ( A. x  -.  x  e.  ( ( A  \  B
)  i^i  C )  ->  ( C  C_  A  ->  C  C_  B )
)
181, 17sylbi 187 1  |-  ( ( ( A  \  B
)  i^i  C )  =  (/)  ->  ( C  C_  A  ->  C  C_  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468
This theorem is referenced by:  tz7.7  4434  tfi  4660  lebnumlem3  18477
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-ne 2461  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3469
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