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Theorem bj-snglex 31096
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 3063 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
2 pweq 3958 . . . . 5  |-  ( x  =  A  ->  ~P x  =  ~P A
)
32eximi 1677 . . . 4  |-  ( E. x  x  =  A  ->  E. x ~P x  =  ~P A )
4 bj-snglss 31093 . . . . . 6  |- sngl  A  C_  ~P A
5 sseq2 3464 . . . . . 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 1677 . . . 4  |-  ( E. x ~P x  =  ~P A  ->  E. xsngl  A 
C_  ~P x )
8 vex 3062 . . . . . . 7  |-  x  e. 
_V
98pwex 4577 . . . . . 6  |-  ~P x  e.  _V
109ssex 4538 . . . . 5  |-  (sngl  A  C_ 
~P x  -> sngl  A  e. 
_V )
1110exlimiv 1743 . . . 4  |-  ( E. xsngl  A  C_  ~P x  -> sngl  A  e.  _V )
123, 7, 113syl 18 . . 3  |-  ( E. x  x  =  A  -> sngl  A  e.  _V )
131, 12sylbi 195 . 2  |-  ( A  e.  _V  -> sngl  A  e. 
_V )
14 bj-snglinv 31095 . . 3  |-  A  =  { y  |  {
y }  e. sngl  A }
15 bj-snsetex 31086 . . 3  |-  (sngl  A  e.  _V  ->  { y  |  { y }  e. sngl  A }  e.  _V )
1614, 15syl5eqel 2494 . 2  |-  (sngl  A  e.  _V  ->  A  e.  _V )
1713, 16impbii 187 1  |-  ( A  e.  _V  <-> sngl  A  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1405   E.wex 1633    e. wcel 1842   {cab 2387   _Vcvv 3059    C_ wss 3414   ~Pcpw 3955   {csn 3972  sngl bj-csngl 31088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-pw 3957  df-sn 3973  df-pr 3975  df-bj-sngl 31089
This theorem is referenced by:  bj-tagex  31110
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