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Theorem snssg 3714
Description: The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.)
Assertion
Ref Expression
snssg  |-  ( A  e.  V  ->  ( A  e.  B  <->  { A }  C_  B ) )

Proof of Theorem snssg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eleq1 2233 . 2  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
2 sneq 3592 . . 3  |-  ( x  =  A  ->  { x }  =  { A } )
32sseq1d 3176 . 2  |-  ( x  =  A  ->  ( { x }  C_  B 
<->  { A }  C_  B ) )
4 vex 2733 . . 3  |-  x  e. 
_V
54snss 3707 . 2  |-  ( x  e.  B  <->  { x }  C_  B )
61, 3, 5vtoclbg 2791 1  |-  ( A  e.  V  ->  ( A  e.  B  <->  { A }  C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1348    e. wcel 2141    C_ wss 3121   {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-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 3587
This theorem is referenced by:  snssi  3722  snssd  3723  prssg  3735  ordtri2orexmid  4505  ordtri2or2exmid  4553  ontri2orexmidim  4554  relsng  4712  fvimacnvi  5607  fvimacnv  5608  strslfv  12447  isneip  12899  elnei  12905  iscnp4  12971  cnpnei  12972
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