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Theorem snsspw 3784
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 3230 . . 3  |-  ( x  =  A  ->  x  C_  A )
2 elsn 3655 . . 3  |-  ( x  e.  { A }  <->  x  =  A )
3 df-pw 3627 . . . 4  |-  ~P A  =  { x  |  x 
C_  A }
43abeq2i 2390 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
51, 2, 43imtr4i 257 . 2  |-  ( x  e.  { A }  ->  x  e.  ~P A
)
65ssriv 3184 1  |-  { A }  C_  ~P A
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   {csn 3640
This theorem is referenced by:  snexALT  4196  snwf  7481  tsksn  8382
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-in 3159  df-ss 3166  df-pw 3627  df-sn 3646
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