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Theorem ssdifeq0 3352
Description: A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.)
Assertion
Ref Expression
ssdifeq0  |-  ( A 
C_  ( B  \  A )  <->  A  =  (/) )

Proof of Theorem ssdifeq0
StepHypRef Expression
1 inidm 3198 . . 3  |-  ( A  i^i  A )  =  A
2 ssdifin0 3351 . . 3  |-  ( A 
C_  ( B  \  A )  ->  ( A  i^i  A )  =  (/) )
31, 2syl5eqr 2131 . 2  |-  ( A 
C_  ( B  \  A )  ->  A  =  (/) )
4 0ss 3309 . . 3  |-  (/)  C_  ( B  \  (/) )
5 id 19 . . . 4  |-  ( A  =  (/)  ->  A  =  (/) )
6 difeq2 3101 . . . 4  |-  ( A  =  (/)  ->  ( B 
\  A )  =  ( B  \  (/) ) )
75, 6sseq12d 3044 . . 3  |-  ( A  =  (/)  ->  ( A 
C_  ( B  \  A )  <->  (/)  C_  ( B  \  (/) ) ) )
84, 7mpbiri 166 . 2  |-  ( A  =  (/)  ->  A  C_  ( B  \  A ) )
93, 8impbii 124 1  |-  ( A 
C_  ( B  \  A )  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1287    \ cdif 2985    i^i cin 2987    C_ wss 2988   (/)c0 3275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rab 2364  df-v 2617  df-dif 2990  df-in 2994  df-ss 3001  df-nul 3276
This theorem is referenced by: (None)
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