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Theorem snssg 3656
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 2202 . 2  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
2 sneq 3538 . . 3  |-  ( x  =  A  ->  { x }  =  { A } )
32sseq1d 3126 . 2  |-  ( x  =  A  ->  ( { x }  C_  B 
<->  { A }  C_  B ) )
4 vex 2689 . . 3  |-  x  e. 
_V
54snss 3649 . 2  |-  ( x  e.  B  <->  { x }  C_  B )
61, 3, 5vtoclbg 2747 1  |-  ( A  e.  V  ->  ( A  e.  B  <->  { A }  C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331    e. wcel 1480    C_ wss 3071   {csn 3527
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 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 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-sn 3533
This theorem is referenced by:  snssi  3664  snssd  3665  prssg  3677  ordtri2orexmid  4438  ordtri2or2exmid  4486  relsng  4642  fvimacnvi  5534  fvimacnv  5535  strslfv  12017  isneip  12329  elnei  12335  iscnp4  12401  cnpnei  12402
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