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Theorem ssdifeq0 3710
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 3550 . . 3  |-  ( A  i^i  A )  =  A
2 ssdifin0 3709 . . 3  |-  ( A 
C_  ( B  \  A )  ->  ( A  i^i  A )  =  (/) )
31, 2syl5eqr 2482 . 2  |-  ( A 
C_  ( B  \  A )  ->  A  =  (/) )
4 0ss 3656 . . 3  |-  (/)  C_  ( B  \  (/) )
5 id 20 . . . 4  |-  ( A  =  (/)  ->  A  =  (/) )
6 difeq2 3459 . . . 4  |-  ( A  =  (/)  ->  ( B 
\  A )  =  ( B  \  (/) ) )
75, 6sseq12d 3377 . . 3  |-  ( A  =  (/)  ->  ( A 
C_  ( B  \  A )  <->  (/)  C_  ( B  \  (/) ) ) )
84, 7mpbiri 225 . 2  |-  ( A  =  (/)  ->  A  C_  ( B  \  A ) )
93, 8impbii 181 1  |-  ( A 
C_  ( B  \  A )  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    \ cdif 3317    i^i cin 3319    C_ wss 3320   (/)c0 3628
This theorem is referenced by:  disjdifprg  24017  measxun2  24564  measssd  24569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rab 2714  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629
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