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Theorem difin0ss 3638
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 3586 . 2  |-  ( ( ( A  \  B
)  i^i  C )  =  (/)  <->  A. x  -.  x  e.  ( ( A  \  B )  i^i  C
) )
2 iman 414 . . . . . 6  |-  ( ( x  e.  C  -> 
( x  e.  A  ->  x  e.  B ) )  <->  -.  ( x  e.  C  /\  -.  (
x  e.  A  ->  x  e.  B )
) )
3 elin 3474 . . . . . . . 8  |-  ( x  e.  ( ( A 
\  B )  i^i 
C )  <->  ( x  e.  ( A  \  B
)  /\  x  e.  C ) )
4 eldif 3274 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
54anbi1i 677 . . . . . . . 8  |-  ( ( x  e.  ( A 
\  B )  /\  x  e.  C )  <->  ( ( x  e.  A  /\  -.  x  e.  B
)  /\  x  e.  C ) )
63, 5bitri 241 . . . . . . 7  |-  ( x  e.  ( ( A 
\  B )  i^i 
C )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  /\  x  e.  C ) )
7 ancom 438 . . . . . . 7  |-  ( ( x  e.  C  /\  ( x  e.  A  /\  -.  x  e.  B
) )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  /\  x  e.  C ) )
8 annim 415 . . . . . . . 8  |-  ( ( x  e.  A  /\  -.  x  e.  B
)  <->  -.  ( x  e.  A  ->  x  e.  B ) )
98anbi2i 676 . . . . . . 7  |-  ( ( x  e.  C  /\  ( x  e.  A  /\  -.  x  e.  B
) )  <->  ( x  e.  C  /\  -.  (
x  e.  A  ->  x  e.  B )
) )
106, 7, 93bitr2i 265 . . . . . 6  |-  ( x  e.  ( ( A 
\  B )  i^i 
C )  <->  ( x  e.  C  /\  -.  (
x  e.  A  ->  x  e.  B )
) )
112, 10xchbinxr 303 . . . . 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 205 . . . 4  |-  ( -.  x  e.  ( ( A  \  B )  i^i  C )  -> 
( ( x  e.  C  ->  x  e.  A )  ->  (
x  e.  C  ->  x  e.  B )
) )
1413al2imi 1567 . . 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 3281 . . 3  |-  ( C 
C_  A  <->  A. x
( x  e.  C  ->  x  e.  A ) )
16 dfss2 3281 . . 3  |-  ( C 
C_  B  <->  A. x
( x  e.  C  ->  x  e.  B ) )
1714, 15, 163imtr4g 262 . 2  |-  ( A. x  -.  x  e.  ( ( A  \  B
)  i^i  C )  ->  ( C  C_  A  ->  C  C_  B )
)
181, 17sylbi 188 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 359   A.wal 1546    = wceq 1649    e. wcel 1717    \ cdif 3261    i^i cin 3263    C_ wss 3264   (/)c0 3572
This theorem is referenced by:  tz7.7  4549  tfi  4774  lebnumlem3  18860
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-v 2902  df-dif 3267  df-in 3271  df-ss 3278  df-nul 3573
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