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Theorem ssdifeq0 3879
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 3672 . . 3  |-  ( A  i^i  A )  =  A
2 ssdifin0 3878 . . 3  |-  ( A 
C_  ( B  \  A )  ->  ( A  i^i  A )  =  (/) )
31, 2syl5eqr 2478 . 2  |-  ( A 
C_  ( B  \  A )  ->  A  =  (/) )
4 0ss 3792 . . 3  |-  (/)  C_  ( B  \  (/) )
5 id 23 . . . 4  |-  ( A  =  (/)  ->  A  =  (/) )
6 difeq2 3578 . . . 4  |-  ( A  =  (/)  ->  ( B 
\  A )  =  ( B  \  (/) ) )
75, 6sseq12d 3494 . . 3  |-  ( A  =  (/)  ->  ( A 
C_  ( B  \  A )  <->  (/)  C_  ( B  \  (/) ) ) )
84, 7mpbiri 237 . 2  |-  ( A  =  (/)  ->  A  C_  ( B  \  A ) )
93, 8impbii 191 1  |-  ( A 
C_  ( B  \  A )  <->  A  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    = wceq 1438    \ cdif 3434    i^i cin 3436    C_ wss 3437   (/)c0 3762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rab 2785  df-v 3084  df-dif 3440  df-in 3444  df-ss 3451  df-nul 3763
This theorem is referenced by:  disjdifprg  28181  measxun2  29034  measssd  29039  pmeasmono  29159
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