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Theorem ssdifsn 3646
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 3199 . . 3  |-  ( A 
C_  ( B  \  { C } )  ->  A  C_  B )
2 reldisj 3409 . . . 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 451 . 2  |-  ( A 
C_  ( B  \  { C } )  <->  ( A  C_  B  /\  ( A  i^i  { C }
)  =  (/) ) )
5 disjsn 3580 . . 3  |-  ( ( A  i^i  { C } )  =  (/)  <->  -.  C  e.  A )
65anbi2i 452 . 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 1331    e. wcel 1480    \ cdif 3063    i^i cin 3065    C_ wss 3066   (/)c0 3358   {csn 3522
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-dif 3068  df-in 3072  df-ss 3079  df-nul 3359  df-sn 3528
This theorem is referenced by: (None)
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