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Theorem snsspw 3962
Description: The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
snsspw  |-  { A }  C_  ~P A

Proof of Theorem snsspw
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqimss 3392 . . 3  |-  ( x  =  A  ->  x  C_  A )
2 elsn 3821 . . 3  |-  ( x  e.  { A }  <->  x  =  A )
3 df-pw 3793 . . . 4  |-  ~P A  =  { x  |  x 
C_  A }
43abeq2i 2542 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
51, 2, 43imtr4i 258 . 2  |-  ( x  e.  { A }  ->  x  e.  ~P A
)
65ssriv 3344 1  |-  { A }  C_  ~P A
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725    C_ wss 3312   ~Pcpw 3791   {csn 3806
This theorem is referenced by:  snexALT  4377  snwf  7725  tsksn  8625
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-in 3319  df-ss 3326  df-pw 3793  df-sn 3812
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