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Theorem ssdifeq0 3413
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 3253 . . 3  |-  ( A  i^i  A )  =  A
2 ssdifin0 3412 . . 3  |-  ( A 
C_  ( B  \  A )  ->  ( A  i^i  A )  =  (/) )
31, 2syl5eqr 2162 . 2  |-  ( A 
C_  ( B  \  A )  ->  A  =  (/) )
4 0ss 3369 . . 3  |-  (/)  C_  ( B  \  (/) )
5 id 19 . . . 4  |-  ( A  =  (/)  ->  A  =  (/) )
6 difeq2 3156 . . . 4  |-  ( A  =  (/)  ->  ( B 
\  A )  =  ( B  \  (/) ) )
75, 6sseq12d 3096 . . 3  |-  ( A  =  (/)  ->  ( A 
C_  ( B  \  A )  <->  (/)  C_  ( B  \  (/) ) ) )
84, 7mpbiri 167 . 2  |-  ( A  =  (/)  ->  A  C_  ( B  \  A ) )
93, 8impbii 125 1  |-  ( A 
C_  ( B  \  A )  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1314    \ cdif 3036    i^i cin 3038    C_ wss 3039   (/)c0 3331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rab 2400  df-v 2660  df-dif 3041  df-in 3045  df-ss 3052  df-nul 3332
This theorem is referenced by: (None)
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