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Theorem sssu 25141
Description: Equality of a class difference and it subtrahend. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
Assertion
Ref Expression
sssu  |-  ( ( B  \  A )  =  A  <->  ( A  =  (/)  /\  B  =  (/) ) )

Proof of Theorem sssu
StepHypRef Expression
1 eqss 3194 . . 3  |-  ( ( B  \  A )  =  A  <->  ( ( B  \  A )  C_  A  /\  A  C_  ( B  \  A ) ) )
2 ssdifeq0 3536 . . . . 5  |-  ( A 
C_  ( B  \  A )  <->  A  =  (/) )
3 ssundif 3537 . . . . . 6  |-  ( B 
C_  ( A  u.  A )  <->  ( B  \  A )  C_  A
)
4 id 19 . . . . . . . . 9  |-  ( A  =  (/)  ->  A  =  (/) )
54, 4uneq12d 3330 . . . . . . . 8  |-  ( A  =  (/)  ->  ( A  u.  A )  =  ( (/)  u.  (/) ) )
6 un0 3479 . . . . . . . 8  |-  ( (/)  u.  (/) )  =  (/)
7 eqtr 2300 . . . . . . . . . 10  |-  ( ( ( A  u.  A
)  =  ( (/)  u.  (/) )  /\  ( (/) 
u.  (/) )  =  (/) )  ->  ( A  u.  A )  =  (/) )
87ex 423 . . . . . . . . 9  |-  ( ( A  u.  A )  =  ( (/)  u.  (/) )  -> 
( ( (/)  u.  (/) )  =  (/)  ->  ( A  u.  A )  =  (/) ) )
9 sseq2 3200 . . . . . . . . . . 11  |-  ( ( A  u.  A )  =  (/)  ->  ( B 
C_  ( A  u.  A )  <->  B  C_  (/) ) )
10 ss0b 3484 . . . . . . . . . . . 12  |-  ( B 
C_  (/)  <->  B  =  (/) )
11 pm3.21 435 . . . . . . . . . . . 12  |-  ( B  =  (/)  ->  ( A  =  (/)  ->  ( A  =  (/)  /\  B  =  (/) ) ) )
1210, 11sylbi 187 . . . . . . . . . . 11  |-  ( B 
C_  (/)  ->  ( A  =  (/)  ->  ( A  =  (/)  /\  B  =  (/) ) ) )
139, 12syl6bi 219 . . . . . . . . . 10  |-  ( ( A  u.  A )  =  (/)  ->  ( B 
C_  ( A  u.  A )  ->  ( A  =  (/)  ->  ( A  =  (/)  /\  B  =  (/) ) ) ) )
1413com23 72 . . . . . . . . 9  |-  ( ( A  u.  A )  =  (/)  ->  ( A  =  (/)  ->  ( B 
C_  ( A  u.  A )  ->  ( A  =  (/)  /\  B  =  (/) ) ) ) )
158, 14syl6 29 . . . . . . . 8  |-  ( ( A  u.  A )  =  ( (/)  u.  (/) )  -> 
( ( (/)  u.  (/) )  =  (/)  ->  ( A  =  (/)  ->  ( B  C_  ( A  u.  A
)  ->  ( A  =  (/)  /\  B  =  (/) ) ) ) ) )
165, 6, 15ee10 1366 . . . . . . 7  |-  ( A  =  (/)  ->  ( A  =  (/)  ->  ( B 
C_  ( A  u.  A )  ->  ( A  =  (/)  /\  B  =  (/) ) ) ) )
1716pm2.43i 43 . . . . . 6  |-  ( A  =  (/)  ->  ( B 
C_  ( A  u.  A )  ->  ( A  =  (/)  /\  B  =  (/) ) ) )
183, 17syl5bir 209 . . . . 5  |-  ( A  =  (/)  ->  ( ( B  \  A ) 
C_  A  ->  ( A  =  (/)  /\  B  =  (/) ) ) )
192, 18sylbi 187 . . . 4  |-  ( A 
C_  ( B  \  A )  ->  (
( B  \  A
)  C_  A  ->  ( A  =  (/)  /\  B  =  (/) ) ) )
2019impcom 419 . . 3  |-  ( ( ( B  \  A
)  C_  A  /\  A  C_  ( B  \  A ) )  -> 
( A  =  (/)  /\  B  =  (/) ) )
211, 20sylbi 187 . 2  |-  ( ( B  \  A )  =  A  ->  ( A  =  (/)  /\  B  =  (/) ) )
22 difeq1 3287 . . . 4  |-  ( B  =  (/)  ->  ( B 
\  A )  =  ( (/)  \  A ) )
23 0dif 3525 . . . 4  |-  ( (/)  \  A )  =  (/)
2422, 23syl6eq 2331 . . 3  |-  ( B  =  (/)  ->  ( B 
\  A )  =  (/) )
25 eqcom 2285 . . . 4  |-  ( A  =  (/)  <->  (/)  =  A )
2625biimpi 186 . . 3  |-  ( A  =  (/)  ->  (/)  =  A )
2724, 26sylan9eqr 2337 . 2  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( B  \  A )  =  A )
2821, 27impbii 180 1  |-  ( ( B  \  A )  =  A  <->  ( A  =  (/)  /\  B  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    \ cdif 3149    u. cun 3150    C_ wss 3152   (/)c0 3455
This theorem is referenced by:  hpd  26169
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456
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