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Theorem snssd 3582
Description: The singleton of an element of a class is a subset of the class (deduction form). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
Hypothesis
Ref Expression
snssd.1  |-  ( ph  ->  A  e.  B )
Assertion
Ref Expression
snssd  |-  ( ph  ->  { A }  C_  B )

Proof of Theorem snssd
StepHypRef Expression
1 snssd.1 . 2  |-  ( ph  ->  A  e.  B )
2 snssg 3573 . . 3  |-  ( A  e.  B  ->  ( A  e.  B  <->  { A }  C_  B ) )
31, 2syl 14 . 2  |-  ( ph  ->  ( A  e.  B  <->  { A }  C_  B
) )
41, 3mpbid 145 1  |-  ( ph  ->  { A }  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    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:  ecinxp  6365  xpdom3m  6548  ac6sfi  6612  undifdc  6632  iunfidisj  6653  fidcenumlemr  6662  en2other2  6820  un0addcl  8704  un0mulcl  8705  fseq1p1m1  9504  fsumge1  10851  phicl2  11464  bj-omtrans  11806
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