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Theorem ssindif0 3508
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 3502 . 2  |-  ( ( A  i^i  ( _V 
\  B ) )  =  (/)  <->  A  C_  ( _V 
\  ( _V  \  B ) ) )
2 ddif 3308 . . 3  |-  ( _V 
\  ( _V  \  B ) )  =  B
32sseq2i 3203 . 2  |-  ( A 
C_  ( _V  \ 
( _V  \  B
) )  <->  A  C_  B
)
41, 3bitr2i 241 1  |-  ( A 
C_  B  <->  ( A  i^i  ( _V  \  B
) )  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455
This theorem is referenced by:  setind  7419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456
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