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Theorem ssindif0im 3348
Description: Subclass implies empty intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)
Assertion
Ref Expression
ssindif0im  |-  ( A 
C_  B  ->  ( A  i^i  ( _V  \  B ) )  =  (/) )

Proof of Theorem ssindif0im
StepHypRef Expression
1 ddifss 3240 . . 3  |-  B  C_  ( _V  \  ( _V  \  B ) )
2 sstr 3036 . . 3  |-  ( ( A  C_  B  /\  B  C_  ( _V  \ 
( _V  \  B
) ) )  ->  A  C_  ( _V  \ 
( _V  \  B
) ) )
31, 2mpan2 417 . 2  |-  ( A 
C_  B  ->  A  C_  ( _V  \  ( _V  \  B ) ) )
4 disj2 3344 . 2  |-  ( ( A  i^i  ( _V 
\  B ) )  =  (/)  <->  A  C_  ( _V 
\  ( _V  \  B ) ) )
53, 4sylibr 133 1  |-  ( A 
C_  B  ->  ( A  i^i  ( _V  \  B ) )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1290   _Vcvv 2622    \ cdif 2999    i^i cin 3001    C_ wss 3002   (/)c0 3289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-v 2624  df-dif 3004  df-in 3008  df-ss 3015  df-nul 3290
This theorem is referenced by: (None)
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