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Theorem ssdifsn 3563
Description: Subset of a set with an element removed. (Contributed by Emmett Weisz, 7-Jul-2021.) (Proof shortened by JJ, 31-May-2022.)
Assertion
Ref Expression
ssdifsn  |-  ( A 
C_  ( B  \  { C } )  <->  ( A  C_  B  /\  -.  C  e.  A ) )

Proof of Theorem ssdifsn
StepHypRef Expression
1 difss2 3126 . . 3  |-  ( A 
C_  ( B  \  { C } )  ->  A  C_  B )
2 reldisj 3331 . . . 4  |-  ( A 
C_  B  ->  (
( A  i^i  { C } )  =  (/)  <->  A  C_  ( B  \  { C } ) ) )
32bicomd 139 . . 3  |-  ( A 
C_  B  ->  ( A  C_  ( B  \  { C } )  <->  ( A  i^i  { C } )  =  (/) ) )
41, 3biadan2 444 . 2  |-  ( A 
C_  ( B  \  { C } )  <->  ( A  C_  B  /\  ( A  i^i  { C }
)  =  (/) ) )
5 disjsn 3499 . . 3  |-  ( ( A  i^i  { C } )  =  (/)  <->  -.  C  e.  A )
65anbi2i 445 . 2  |-  ( ( A  C_  B  /\  ( A  i^i  { C } )  =  (/) ) 
<->  ( A  C_  B  /\  -.  C  e.  A
) )
74, 6bitri 182 1  |-  ( A 
C_  ( B  \  { C } )  <->  ( A  C_  B  /\  -.  C  e.  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438    \ cdif 2994    i^i cin 2996    C_ wss 2997   (/)c0 3284   {csn 3441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-dif 2999  df-in 3003  df-ss 3010  df-nul 3285  df-sn 3447
This theorem is referenced by: (None)
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