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Theorem ssdifsn 3709
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 3255 . . 3  |-  ( A 
C_  ( B  \  { C } )  ->  A  C_  B )
2 reldisj 3465 . . . 4  |-  ( A 
C_  B  ->  (
( A  i^i  { C } )  =  (/)  <->  A  C_  ( B  \  { C } ) ) )
32bicomd 140 . . 3  |-  ( A 
C_  B  ->  ( A  C_  ( B  \  { C } )  <->  ( A  i^i  { C } )  =  (/) ) )
41, 3biadan2 453 . 2  |-  ( A 
C_  ( B  \  { C } )  <->  ( A  C_  B  /\  ( A  i^i  { C }
)  =  (/) ) )
5 disjsn 3643 . . 3  |-  ( ( A  i^i  { C } )  =  (/)  <->  -.  C  e.  A )
65anbi2i 454 . 2  |-  ( ( A  C_  B  /\  ( A  i^i  { C } )  =  (/) ) 
<->  ( A  C_  B  /\  -.  C  e.  A
) )
74, 6bitri 183 1  |-  ( A 
C_  ( B  \  { C } )  <->  ( A  C_  B  /\  -.  C  e.  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141    \ cdif 3118    i^i cin 3120    C_ wss 3121   (/)c0 3414   {csn 3581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3587
This theorem is referenced by: (None)
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