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Theorem ssindif0 3823
Description: Subclass expressed in terms of intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)
Assertion
Ref Expression
ssindif0  |-  ( A 
C_  B  <->  ( A  i^i  ( _V  \  B
) )  =  (/) )

Proof of Theorem ssindif0
StepHypRef Expression
1 disj2 3817 . 2  |-  ( ( A  i^i  ( _V 
\  B ) )  =  (/)  <->  A  C_  ( _V 
\  ( _V  \  B ) ) )
2 ddif 3575 . . 3  |-  ( _V 
\  ( _V  \  B ) )  =  B
32sseq2i 3467 . 2  |-  ( A 
C_  ( _V  \ 
( _V  \  B
) )  <->  A  C_  B
)
41, 3bitr2i 250 1  |-  ( A 
C_  B  <->  ( A  i^i  ( _V  \  B
) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1405   _Vcvv 3059    \ cdif 3411    i^i cin 3413    C_ wss 3414   (/)c0 3738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-v 3061  df-dif 3417  df-in 3421  df-ss 3428  df-nul 3739
This theorem is referenced by:  setind  8197
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