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Theorem ssindif0 3832
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 3826 . 2  |-  ( ( A  i^i  ( _V 
\  B ) )  =  (/)  <->  A  C_  ( _V 
\  ( _V  \  B ) ) )
2 ddif 3588 . . 3  |-  ( _V 
\  ( _V  \  B ) )  =  B
32sseq2i 3481 . 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 1370   _Vcvv 3070    \ cdif 3425    i^i cin 3427    C_ wss 3428   (/)c0 3737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-v 3072  df-dif 3431  df-in 3435  df-ss 3442  df-nul 3738
This theorem is referenced by:  setind  8057
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