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Theorem snsspw 3751
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 3201 . . 3  |-  ( x  =  A  ->  x  C_  A )
2 velsn 3600 . . 3  |-  ( x  e.  { A }  <->  x  =  A )
3 df-pw 3568 . . . 4  |-  ~P A  =  { x  |  x 
C_  A }
43abeq2i 2281 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
51, 2, 43imtr4i 200 . 2  |-  ( x  e.  { A }  ->  x  e.  ~P A
)
65ssriv 3151 1  |-  { A }  C_  ~P A
Colors of variables: wff set class
Syntax hints:    = wceq 1348    e. wcel 2141    C_ wss 3121   ~Pcpw 3566   {csn 3583
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589
This theorem is referenced by:  snexg  4170
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