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Theorem bj-snglss 33609
Description: The singletonization of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-snglss  |- sngl  A  C_  ~P A

Proof of Theorem bj-snglss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-elsngl 33607 . . . . 5  |-  ( x  e. sngl  A  <->  E. y  e.  A  x  =  { y } )
2 df-rex 2820 . . . . . 6  |-  ( E. y  e.  A  x  =  { y }  <->  E. y ( y  e.  A  /\  x  =  { y } ) )
3 snssi 4171 . . . . . . . 8  |-  ( y  e.  A  ->  { y }  C_  A )
4 sseq1 3525 . . . . . . . . 9  |-  ( x  =  { y }  ->  ( x  C_  A 
<->  { y }  C_  A ) )
54biimparc 487 . . . . . . . 8  |-  ( ( { y }  C_  A  /\  x  =  {
y } )  ->  x  C_  A )
63, 5sylan 471 . . . . . . 7  |-  ( ( y  e.  A  /\  x  =  { y } )  ->  x  C_  A )
76eximi 1635 . . . . . 6  |-  ( E. y ( y  e.  A  /\  x  =  { y } )  ->  E. y  x  C_  A )
82, 7sylbi 195 . . . . 5  |-  ( E. y  e.  A  x  =  { y }  ->  E. y  x  C_  A )
91, 8sylbi 195 . . . 4  |-  ( x  e. sngl  A  ->  E. y  x  C_  A )
10 ax5e 1682 . . . 4  |-  ( E. y  x  C_  A  ->  x  C_  A )
119, 10syl 16 . . 3  |-  ( x  e. sngl  A  ->  x  C_  A )
12 selpw 4017 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
1311, 12sylibr 212 . 2  |-  ( x  e. sngl  A  ->  x  e.  ~P A )
1413ssriv 3508 1  |- sngl  A  C_  ~P A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   E.wrex 2815    C_ wss 3476   ~Pcpw 4010   {csn 4027  sngl bj-csngl 33604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-pw 4012  df-sn 4028  df-pr 4030  df-bj-sngl 33605
This theorem is referenced by:  bj-snglex  33612  bj-tagss  33619
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