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Theorem ssdifeq0 3570
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 3412 . . 3  |-  ( A  i^i  A )  =  A
2 ssdifin0 3569 . . 3  |-  ( A 
C_  ( B  \  A )  ->  ( A  i^i  A )  =  (/) )
31, 2syl5eqr 2362 . 2  |-  ( A 
C_  ( B  \  A )  ->  A  =  (/) )
4 0ss 3517 . . 3  |-  (/)  C_  ( B  \  (/) )
5 id 19 . . . 4  |-  ( A  =  (/)  ->  A  =  (/) )
6 difeq2 3322 . . . 4  |-  ( A  =  (/)  ->  ( B 
\  A )  =  ( B  \  (/) ) )
75, 6sseq12d 3241 . . 3  |-  ( A  =  (/)  ->  ( A 
C_  ( B  \  A )  <->  (/)  C_  ( B  \  (/) ) ) )
84, 7mpbiri 224 . 2  |-  ( A  =  (/)  ->  A  C_  ( B  \  A ) )
93, 8impbii 180 1  |-  ( A 
C_  ( B  \  A )  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1633    \ cdif 3183    i^i cin 3185    C_ wss 3186   (/)c0 3489
This theorem is referenced by:  disjdifprg  23160  measxun2  23738  measssd  23743
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rab 2586  df-v 2824  df-dif 3189  df-in 3193  df-ss 3200  df-nul 3490
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