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Theorem bj-snglss 34948
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 34946 . . . . 5  |-  ( x  e. sngl  A  <->  E. y  e.  A  x  =  { y } )
2 df-rex 2810 . . . . . 6  |-  ( E. y  e.  A  x  =  { y }  <->  E. y ( y  e.  A  /\  x  =  { y } ) )
3 snssi 4160 . . . . . . . 8  |-  ( y  e.  A  ->  { y }  C_  A )
4 sseq1 3510 . . . . . . . . 9  |-  ( x  =  { y }  ->  ( x  C_  A 
<->  { y }  C_  A ) )
54biimparc 485 . . . . . . . 8  |-  ( ( { y }  C_  A  /\  x  =  {
y } )  ->  x  C_  A )
63, 5sylan 469 . . . . . . 7  |-  ( ( y  e.  A  /\  x  =  { y } )  ->  x  C_  A )
76eximi 1661 . . . . . 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 1711 . . . 4  |-  ( E. y  x  C_  A  ->  x  C_  A )
119, 10syl 16 . . 3  |-  ( x  e. sngl  A  ->  x  C_  A )
12 selpw 4006 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
1311, 12sylibr 212 . 2  |-  ( x  e. sngl  A  ->  x  e.  ~P A )
1413ssriv 3493 1  |- sngl  A  C_  ~P A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823   E.wrex 2805    C_ wss 3461   ~Pcpw 3999   {csn 4016  sngl bj-csngl 34943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-pw 4001  df-sn 4017  df-pr 4019  df-bj-sngl 34944
This theorem is referenced by:  bj-snglex  34951  bj-tagss  34958
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