ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  snsspw Unicode version

Theorem snsspw 3659
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 3119 . . 3  |-  ( x  =  A  ->  x  C_  A )
2 velsn 3512 . . 3  |-  ( x  e.  { A }  <->  x  =  A )
3 df-pw 3480 . . . 4  |-  ~P A  =  { x  |  x 
C_  A }
43abeq2i 2226 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
51, 2, 43imtr4i 200 . 2  |-  ( x  e.  { A }  ->  x  e.  ~P A
)
65ssriv 3069 1  |-  { A }  C_  ~P A
Colors of variables: wff set class
Syntax hints:    = wceq 1314    e. wcel 1463    C_ wss 3039   ~Pcpw 3478   {csn 3495
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501
This theorem is referenced by:  snexg  4076
  Copyright terms: Public domain W3C validator