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Theorem bj-snglex 32748
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 3058 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
2 pweq 3947 . . . . 5  |-  ( x  =  A  ->  ~P x  =  ~P A
)
32eximi 1626 . . . 4  |-  ( E. x  x  =  A  ->  E. x ~P x  =  ~P A )
4 bj-snglss 32745 . . . . . 6  |- sngl  A  C_  ~P A
5 sseq2 3462 . . . . . 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 1626 . . . 4  |-  ( E. x ~P x  =  ~P A  ->  E. xsngl  A 
C_  ~P x )
8 vex 3057 . . . . . . 7  |-  x  e. 
_V
98pwex 4559 . . . . . 6  |-  ~P x  e.  _V
109ssex 4520 . . . . 5  |-  (sngl  A  C_ 
~P x  -> sngl  A  e. 
_V )
1110exlimiv 1689 . . . 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 32747 . . 3  |-  A  =  { y  |  {
y }  e. sngl  A }
15 bj-snsetex 32738 . . 3  |-  (sngl  A  e.  _V  ->  { y  |  { y }  e. sngl  A }  e.  _V )
1614, 15syl5eqel 2540 . 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 1370   E.wex 1587    e. wcel 1757   {cab 2435   _Vcvv 3054    C_ wss 3412   ~Pcpw 3944   {csn 3961  sngl bj-csngl 32740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-rex 2798  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-pw 3946  df-sn 3962  df-pr 3964  df-bj-sngl 32741
This theorem is referenced by:  bj-tagex  32762
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