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Theorem snssd 3551
Description: The singleton of an element of a class is a subset of the class (deduction rule). (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 3542 . . 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 1434    C_ wss 2983   {csn 3417
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2613  df-in 2989  df-ss 2996  df-sn 3423
This theorem is referenced by:  ecinxp  6270  xpdom3m  6401  ac6sfi  6456  undifdc  6470  en2other2  6599  un0addcl  8465  un0mulcl  8466  fseq1p1m1  9264  phicl2  10822  bj-omtrans  11043
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