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Theorem sssnm 3649
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 3059 . . . . . . . . . 10  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  x  e.  { B } ) )
2 elsni 3513 . . . . . . . . . 10  |-  ( x  e.  { B }  ->  x  =  B )
31, 2syl6 33 . . . . . . . . 9  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  x  =  B ) )
4 eleq1 2178 . . . . . . . . 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 177 . . . . . . 7  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  B  e.  A ) )
76exlimdv 1773 . . . . . 6  |-  ( A 
C_  { B }  ->  ( E. x  x  e.  A  ->  B  e.  A ) )
8 snssi 3632 . . . . . 6  |-  ( B  e.  A  ->  { B }  C_  A )
97, 8syl6 33 . . . . 5  |-  ( A 
C_  { B }  ->  ( E. x  x  e.  A  ->  { B }  C_  A ) )
109anc2li 325 . . . 4  |-  ( A 
C_  { B }  ->  ( E. x  x  e.  A  ->  ( A  C_  { B }  /\  { B }  C_  A ) ) )
11 eqss 3080 . . . 4  |-  ( A  =  { B }  <->  ( A  C_  { B }  /\  { B }  C_  A ) )
1210, 11syl6ibr 161 . . 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 3119 . 2  |-  ( A  =  { B }  ->  A  C_  { B } )
1513, 14impbid1 141 1  |-  ( E. x  x  e.  A  ->  ( A  C_  { B } 
<->  A  =  { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314   E.wex 1451    e. wcel 1463    C_ wss 3039   {csn 3495
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-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045  df-ss 3052  df-sn 3501
This theorem is referenced by:  eqsnm  3650  exmid01  4089  exmidn0m  4092  exmidsssn  4093  exmidomni  6980  exmidunben  11834  exmidsbthrlem  13019  sbthomlem  13022
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