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Theorem ssdifsn 3735
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 3278 . . 3  |-  ( A 
C_  ( B  \  { C } )  ->  A  C_  B )
2 reldisj 3489 . . . 4  |-  ( A 
C_  B  ->  (
( A  i^i  { C } )  =  (/)  <->  A  C_  ( B  \  { C } ) ) )
32bicomd 141 . . 3  |-  ( A 
C_  B  ->  ( A  C_  ( B  \  { C } )  <->  ( A  i^i  { C } )  =  (/) ) )
41, 3biadan2 456 . 2  |-  ( A 
C_  ( B  \  { C } )  <->  ( A  C_  B  /\  ( A  i^i  { C }
)  =  (/) ) )
5 disjsn 3669 . . 3  |-  ( ( A  i^i  { C } )  =  (/)  <->  -.  C  e.  A )
65anbi2i 457 . 2  |-  ( ( A  C_  B  /\  ( A  i^i  { C } )  =  (/) ) 
<->  ( A  C_  B  /\  -.  C  e.  A
) )
74, 6bitri 184 1  |-  ( A 
C_  ( B  \  { C } )  <->  ( A  C_  B  /\  -.  C  e.  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160    \ cdif 3141    i^i cin 3143    C_ wss 3144   (/)c0 3437   {csn 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438  df-sn 3613
This theorem is referenced by: (None)
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