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Theorem snsspw 3727
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 3182 . . 3  |-  ( x  =  A  ->  x  C_  A )
2 velsn 3577 . . 3  |-  ( x  e.  { A }  <->  x  =  A )
3 df-pw 3545 . . . 4  |-  ~P A  =  { x  |  x 
C_  A }
43abeq2i 2268 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
51, 2, 43imtr4i 200 . 2  |-  ( x  e.  { A }  ->  x  e.  ~P A
)
65ssriv 3132 1  |-  { A }  C_  ~P A
Colors of variables: wff set class
Syntax hints:    = wceq 1335    e. wcel 2128    C_ wss 3102   ~Pcpw 3543   {csn 3560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566
This theorem is referenced by:  snexg  4145
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