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Theorem snssd 3506
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 3497 . . 3  |-  ( A  e.  B  ->  ( A  e.  B  <->  { A }  C_  B ) )
31, 2syl 14 . 2  |-  ( ph  ->  ( A  e.  B  <->  { A }  C_  B
) )
41, 3mpbid 135 1  |-  ( ph  ->  { A }  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98    e. wcel 1393    C_ wss 2914   {csn 3372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-in 2921  df-ss 2928  df-sn 3378
This theorem is referenced by:  ecinxp  6168  xpdom3m  6295  ac6sfi  6338  un0addcl  8187  un0mulcl  8188  fseq1p1m1  8923  bj-omtrans  9954
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