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Theorem bj-snglex 31560
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 3048 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
2 pweq 3953 . . . . 5  |-  ( x  =  A  ->  ~P x  =  ~P A
)
32eximi 1706 . . . 4  |-  ( E. x  x  =  A  ->  E. x ~P x  =  ~P A )
4 bj-snglss 31557 . . . . . 6  |- sngl  A  C_  ~P A
5 sseq2 3453 . . . . . 6  |-  ( ~P x  =  ~P A  ->  (sngl  A  C_  ~P x 
<-> sngl 
A  C_  ~P A
) )
64, 5mpbiri 237 . . . . 5  |-  ( ~P x  =  ~P A  -> sngl  A  C_  ~P x
)
76eximi 1706 . . . 4  |-  ( E. x ~P x  =  ~P A  ->  E. xsngl  A 
C_  ~P x )
8 vex 3047 . . . . . . 7  |-  x  e. 
_V
98pwex 4585 . . . . . 6  |-  ~P x  e.  _V
109ssex 4546 . . . . 5  |-  (sngl  A  C_ 
~P x  -> sngl  A  e. 
_V )
1110exlimiv 1775 . . . 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 199 . 2  |-  ( A  e.  _V  -> sngl  A  e. 
_V )
14 bj-snglinv 31559 . . 3  |-  A  =  { y  |  {
y }  e. sngl  A }
15 bj-snsetex 31550 . . 3  |-  (sngl  A  e.  _V  ->  { y  |  { y }  e. sngl  A }  e.  _V )
1614, 15syl5eqel 2532 . 2  |-  (sngl  A  e.  _V  ->  A  e.  _V )
1713, 16impbii 191 1  |-  ( A  e.  _V  <-> sngl  A  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    = wceq 1443   E.wex 1662    e. wcel 1886   {cab 2436   _Vcvv 3044    C_ wss 3403   ~Pcpw 3950   {csn 3967  sngl bj-csngl 31552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-pw 3952  df-sn 3968  df-pr 3970  df-bj-sngl 31553
This theorem is referenced by:  bj-tagex  31574
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