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Theorem ssdifeq0 3476
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 3316 . . 3  |-  ( A  i^i  A )  =  A
2 ssdifin0 3475 . . 3  |-  ( A 
C_  ( B  \  A )  ->  ( A  i^i  A )  =  (/) )
31, 2eqtr3id 2204 . 2  |-  ( A 
C_  ( B  \  A )  ->  A  =  (/) )
4 0ss 3432 . . 3  |-  (/)  C_  ( B  \  (/) )
5 id 19 . . . 4  |-  ( A  =  (/)  ->  A  =  (/) )
6 difeq2 3219 . . . 4  |-  ( A  =  (/)  ->  ( B 
\  A )  =  ( B  \  (/) ) )
75, 6sseq12d 3159 . . 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 1335    \ cdif 3099    i^i cin 3101    C_ wss 3102   (/)c0 3394
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rab 2444  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3395
This theorem is referenced by: (None)
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