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Theorem snsspw 3912
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 3343 . . 3  |-  ( x  =  A  ->  x  C_  A )
2 elsn 3772 . . 3  |-  ( x  e.  { A }  <->  x  =  A )
3 df-pw 3744 . . . 4  |-  ~P A  =  { x  |  x 
C_  A }
43abeq2i 2494 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
51, 2, 43imtr4i 258 . 2  |-  ( x  e.  { A }  ->  x  e.  ~P A
)
65ssriv 3295 1  |-  { A }  C_  ~P A
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717    C_ wss 3263   ~Pcpw 3742   {csn 3757
This theorem is referenced by:  snexALT  4326  snwf  7668  tsksn  8568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-in 3270  df-ss 3277  df-pw 3744  df-sn 3763
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