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Theorem sssnm 3598
Description: The inhabited subset of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)
Assertion
Ref Expression
sssnm  |-  ( E. x  x  e.  A  ->  ( A  C_  { B } 
<->  A  =  { B } ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem sssnm
StepHypRef Expression
1 ssel 3019 . . . . . . . . . 10  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  x  e.  { B } ) )
2 elsni 3464 . . . . . . . . . 10  |-  ( x  e.  { B }  ->  x  =  B )
31, 2syl6 33 . . . . . . . . 9  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  x  =  B ) )
4 eleq1 2150 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
53, 4syl6 33 . . . . . . . 8  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  ( x  e.  A  <->  B  e.  A ) ) )
65ibd 176 . . . . . . 7  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  B  e.  A ) )
76exlimdv 1747 . . . . . 6  |-  ( A 
C_  { B }  ->  ( E. x  x  e.  A  ->  B  e.  A ) )
8 snssi 3581 . . . . . 6  |-  ( B  e.  A  ->  { B }  C_  A )
97, 8syl6 33 . . . . 5  |-  ( A 
C_  { B }  ->  ( E. x  x  e.  A  ->  { B }  C_  A ) )
109anc2li 322 . . . 4  |-  ( A 
C_  { B }  ->  ( E. x  x  e.  A  ->  ( A  C_  { B }  /\  { B }  C_  A ) ) )
11 eqss 3040 . . . 4  |-  ( A  =  { B }  <->  ( A  C_  { B }  /\  { B }  C_  A ) )
1210, 11syl6ibr 160 . . 3  |-  ( A 
C_  { B }  ->  ( E. x  x  e.  A  ->  A  =  { B } ) )
1312com12 30 . 2  |-  ( E. x  x  e.  A  ->  ( A  C_  { B }  ->  A  =  { B } ) )
14 eqimss 3078 . 2  |-  ( A  =  { B }  ->  A  C_  { B } )
1513, 14impbid1 140 1  |-  ( E. x  x  e.  A  ->  ( A  C_  { B } 
<->  A  =  { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438    C_ wss 2999   {csn 3446
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-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-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3005  df-ss 3012  df-sn 3452
This theorem is referenced by:  eqsnm  3599  exmid01  4032  exmidomni  6798  exmidsbthrlem  11912
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