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Theorem snssd 3635
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 3626 . . 3  |-  ( A  e.  B  ->  ( A  e.  B  <->  { A }  C_  B ) )
31, 2syl 14 . 2  |-  ( ph  ->  ( A  e.  B  <->  { A }  C_  B
) )
41, 3mpbid 146 1  |-  ( ph  ->  { A }  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    e. wcel 1465    C_ wss 3041   {csn 3497
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-sn 3503
This theorem is referenced by:  pwntru  4092  ecinxp  6472  xpdom3m  6696  ac6sfi  6760  undifdc  6780  iunfidisj  6802  fidcenumlemr  6811  ssfii  6830  en2other2  7020  un0addcl  8978  un0mulcl  8979  fseq1p1m1  9842  fsumge1  11198  phicl2  11817  ennnfonelemhf1o  11853  rest0  12275  iscnp4  12314  cnconst2  12329  cnpdis  12338  txdis  12373  txdis1cn  12374  fsumcncntop  12652  dvef  12783  bj-omtrans  13081  pwtrufal  13119
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