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Theorem bj-snglex 33612
Description: A class is a set if and only if its singletonization is a set. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-snglex  |-  ( A  e.  _V  <-> sngl  A  e.  _V )

Proof of Theorem bj-snglex
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isset 3117 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
2 pweq 4013 . . . . 5  |-  ( x  =  A  ->  ~P x  =  ~P A
)
32eximi 1635 . . . 4  |-  ( E. x  x  =  A  ->  E. x ~P x  =  ~P A )
4 bj-snglss 33609 . . . . . 6  |- sngl  A  C_  ~P A
5 sseq2 3526 . . . . . 6  |-  ( ~P x  =  ~P A  ->  (sngl  A  C_  ~P x 
<-> sngl 
A  C_  ~P A
) )
64, 5mpbiri 233 . . . . 5  |-  ( ~P x  =  ~P A  -> sngl  A  C_  ~P x
)
76eximi 1635 . . . 4  |-  ( E. x ~P x  =  ~P A  ->  E. xsngl  A 
C_  ~P x )
8 vex 3116 . . . . . . 7  |-  x  e. 
_V
98pwex 4630 . . . . . 6  |-  ~P x  e.  _V
109ssex 4591 . . . . 5  |-  (sngl  A  C_ 
~P x  -> sngl  A  e. 
_V )
1110exlimiv 1698 . . . 4  |-  ( E. xsngl  A  C_  ~P x  -> sngl  A  e.  _V )
123, 7, 113syl 20 . . 3  |-  ( E. x  x  =  A  -> sngl  A  e.  _V )
131, 12sylbi 195 . 2  |-  ( A  e.  _V  -> sngl  A  e. 
_V )
14 bj-snglinv 33611 . . 3  |-  A  =  { y  |  {
y }  e. sngl  A }
15 bj-snsetex 33602 . . 3  |-  (sngl  A  e.  _V  ->  { y  |  { y }  e. sngl  A }  e.  _V )
1614, 15syl5eqel 2559 . 2  |-  (sngl  A  e.  _V  ->  A  e.  _V )
1713, 16impbii 188 1  |-  ( A  e.  _V  <-> sngl  A  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452   _Vcvv 3113    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-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  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-sbc 3332  df-csb 3436  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-tagex  33626
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