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Theorem sssnm 3741
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 3141 . . . . . . . . . 10  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  x  e.  { B } ) )
2 elsni 3601 . . . . . . . . . 10  |-  ( x  e.  { B }  ->  x  =  B )
31, 2syl6 33 . . . . . . . . 9  |-  ( A 
C_  { B }  ->  ( x  e.  A  ->  x  =  B ) )
4 eleq1 2233 . . . . . . . . 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 1812 . . . . . 6  |-  ( A 
C_  { B }  ->  ( E. x  x  e.  A  ->  B  e.  A ) )
8 snssi 3724 . . . . . 6  |-  ( B  e.  A  ->  { B }  C_  A )
97, 8syl6 33 . . . . 5  |-  ( A 
C_  { B }  ->  ( E. x  x  e.  A  ->  { B }  C_  A ) )
109anc2li 327 . . . 4  |-  ( A 
C_  { B }  ->  ( E. x  x  e.  A  ->  ( A  C_  { B }  /\  { B }  C_  A ) ) )
11 eqss 3162 . . . 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 3201 . 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 1348   E.wex 1485    e. wcel 2141    C_ wss 3121   {csn 3583
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 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-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-sn 3589
This theorem is referenced by:  eqsnm  3742  ss1o0el1  4183  exmidn0m  4187  exmidsssn  4188  exmidomni  7118  exmidunben  12381  exmidsbthrlem  14054  sbthomlem  14057
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