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Theorem snsspw 3800
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 3243 . . 3  |-  ( x  =  A  ->  x  C_  A )
2 elsn 3668 . . 3  |-  ( x  e.  { A }  <->  x  =  A )
3 df-pw 3640 . . . 4  |-  ~P A  =  { x  |  x 
C_  A }
43abeq2i 2403 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
51, 2, 43imtr4i 257 . 2  |-  ( x  e.  { A }  ->  x  e.  ~P A
)
65ssriv 3197 1  |-  { A }  C_  ~P A
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696    C_ wss 3165   ~Pcpw 3638   {csn 3653
This theorem is referenced by:  snexALT  4212  snwf  7497  tsksn  8398
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-in 3172  df-ss 3179  df-pw 3640  df-sn 3659
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