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Theorem ssindif0 3681
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 3675 . 2  |-  ( ( A  i^i  ( _V 
\  B ) )  =  (/)  <->  A  C_  ( _V 
\  ( _V  \  B ) ) )
2 ddif 3479 . . 3  |-  ( _V 
\  ( _V  \  B ) )  =  B
32sseq2i 3373 . 2  |-  ( A 
C_  ( _V  \ 
( _V  \  B
) )  <->  A  C_  B
)
41, 3bitr2i 242 1  |-  ( A 
C_  B  <->  ( A  i^i  ( _V  \  B
) )  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652   _Vcvv 2956    \ cdif 3317    i^i cin 3319    C_ wss 3320   (/)c0 3628
This theorem is referenced by:  setind  7673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629
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