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Theorem bj-snglc 34875
Description: Characterization of the elements of  A in terms of elements of its singletonization. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-snglc  |-  ( A  e.  B  <->  { A }  e. sngl  B )

Proof of Theorem bj-snglc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-rex 2738 . 2  |-  ( E. x  e.  B  { A }  =  {
x }  <->  E. x
( x  e.  B  /\  { A }  =  { x } ) )
2 bj-elsngl 34874 . 2  |-  ( { A }  e. sngl  B  <->  E. x  e.  B  { A }  =  {
x } )
3 elisset 3045 . . . . 5  |-  ( A  e.  B  ->  E. x  x  =  A )
43pm4.71i 630 . . . 4  |-  ( A  e.  B  <->  ( A  e.  B  /\  E. x  x  =  A )
)
5 19.42v 1783 . . . 4  |-  ( E. x ( A  e.  B  /\  x  =  A )  <->  ( A  e.  B  /\  E. x  x  =  A )
)
6 eleq1 2454 . . . . . . 7  |-  ( A  =  x  ->  ( A  e.  B  <->  x  e.  B ) )
76eqcoms 2394 . . . . . 6  |-  ( x  =  A  ->  ( A  e.  B  <->  x  e.  B ) )
87pm5.32ri 636 . . . . 5  |-  ( ( A  e.  B  /\  x  =  A )  <->  ( x  e.  B  /\  x  =  A )
)
98exbii 1675 . . . 4  |-  ( E. x ( A  e.  B  /\  x  =  A )  <->  E. x
( x  e.  B  /\  x  =  A
) )
104, 5, 93bitr2i 273 . . 3  |-  ( A  e.  B  <->  E. x
( x  e.  B  /\  x  =  A
) )
11 vex 3037 . . . . . . 7  |-  x  e. 
_V
12 sneqbg 4114 . . . . . . 7  |-  ( x  e.  _V  ->  ( { x }  =  { A }  <->  x  =  A ) )
1311, 12ax-mp 5 . . . . . 6  |-  ( { x }  =  { A }  <->  x  =  A
)
14 eqcom 2391 . . . . . 6  |-  ( { x }  =  { A }  <->  { A }  =  { x } )
1513, 14bitr3i 251 . . . . 5  |-  ( x  =  A  <->  { A }  =  { x } )
1615anbi2i 692 . . . 4  |-  ( ( x  e.  B  /\  x  =  A )  <->  ( x  e.  B  /\  { A }  =  {
x } ) )
1716exbii 1675 . . 3  |-  ( E. x ( x  e.  B  /\  x  =  A )  <->  E. x
( x  e.  B  /\  { A }  =  { x } ) )
1810, 17bitri 249 . 2  |-  ( A  e.  B  <->  E. x
( x  e.  B  /\  { A }  =  { x } ) )
191, 2, 183bitr4ri 278 1  |-  ( A  e.  B  <->  { A }  e. sngl  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1399   E.wex 1620    e. wcel 1826   E.wrex 2733   _Vcvv 3034   {csn 3944  sngl bj-csngl 34871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-v 3036  df-dif 3392  df-un 3394  df-nul 3712  df-sn 3945  df-pr 3947  df-bj-sngl 34872
This theorem is referenced by:  bj-snglinv  34878  bj-tagci  34890  bj-tagcg  34891
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