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Theorem snssd 3631
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 3622 . . 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 1463    C_ wss 3037   {csn 3493
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-in 3043  df-ss 3050  df-sn 3499
This theorem is referenced by:  pwntru  4082  ecinxp  6458  xpdom3m  6681  ac6sfi  6745  undifdc  6765  iunfidisj  6786  fidcenumlemr  6795  ssfii  6814  en2other2  7000  un0addcl  8914  un0mulcl  8915  fseq1p1m1  9767  fsumge1  11122  phicl2  11735  ennnfonelemhf1o  11771  rest0  12191  iscnp4  12229  cnconst2  12244  cnpdis  12253  txdis  12288  txdis1cn  12289  fsumcncntop  12542  bj-omtrans  12846  pwtrufal  12884
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