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Theorem difin0ss 3833
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 3747 . 2  |-  ( ( ( A  \  B
)  i^i  C )  =  (/)  <->  A. x  -.  x  e.  ( ( A  \  B )  i^i  C
) )
2 iman 426 . . . . . 6  |-  ( ( x  e.  C  -> 
( x  e.  A  ->  x  e.  B ) )  <->  -.  ( x  e.  C  /\  -.  (
x  e.  A  ->  x  e.  B )
) )
3 elin 3617 . . . . . . . 8  |-  ( x  e.  ( ( A 
\  B )  i^i 
C )  <->  ( x  e.  ( A  \  B
)  /\  x  e.  C ) )
4 eldif 3414 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
54anbi1i 701 . . . . . . . 8  |-  ( ( x  e.  ( A 
\  B )  /\  x  e.  C )  <->  ( ( x  e.  A  /\  -.  x  e.  B
)  /\  x  e.  C ) )
63, 5bitri 253 . . . . . . 7  |-  ( x  e.  ( ( A 
\  B )  i^i 
C )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  /\  x  e.  C ) )
7 ancom 452 . . . . . . 7  |-  ( ( x  e.  C  /\  ( x  e.  A  /\  -.  x  e.  B
) )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  /\  x  e.  C ) )
8 annim 427 . . . . . . . 8  |-  ( ( x  e.  A  /\  -.  x  e.  B
)  <->  -.  ( x  e.  A  ->  x  e.  B ) )
98anbi2i 700 . . . . . . 7  |-  ( ( x  e.  C  /\  ( x  e.  A  /\  -.  x  e.  B
) )  <->  ( x  e.  C  /\  -.  (
x  e.  A  ->  x  e.  B )
) )
106, 7, 93bitr2i 277 . . . . . 6  |-  ( x  e.  ( ( A 
\  B )  i^i 
C )  <->  ( x  e.  C  /\  -.  (
x  e.  A  ->  x  e.  B )
) )
112, 10xchbinxr 313 . . . . 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 217 . . . 4  |-  ( -.  x  e.  ( ( A  \  B )  i^i  C )  -> 
( ( x  e.  C  ->  x  e.  A )  ->  (
x  e.  C  ->  x  e.  B )
) )
1413al2imi 1687 . . 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 3421 . . 3  |-  ( C 
C_  A  <->  A. x
( x  e.  C  ->  x  e.  A ) )
16 dfss2 3421 . . 3  |-  ( C 
C_  B  <->  A. x
( x  e.  C  ->  x  e.  B ) )
1714, 15, 163imtr4g 274 . 2  |-  ( A. x  -.  x  e.  ( ( A  \  B
)  i^i  C )  ->  ( C  C_  A  ->  C  C_  B )
)
181, 17sylbi 199 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 371   A.wal 1442    = wceq 1444    e. wcel 1887    \ cdif 3401    i^i cin 3403    C_ wss 3404   (/)c0 3731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-v 3047  df-dif 3407  df-in 3411  df-ss 3418  df-nul 3732
This theorem is referenced by:  tz7.7  5449  tfi  6680  lebnumlem3  21991  lebnumlem3OLD  21994
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