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Theorem bj-snglc 31531
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 2777 . 2  |-  ( E. x  e.  B  { A }  =  {
x }  <->  E. x
( x  e.  B  /\  { A }  =  { x } ) )
2 bj-elsngl 31530 . 2  |-  ( { A }  e. sngl  B  <->  E. x  e.  B  { A }  =  {
x } )
3 elisset 3091 . . . . 5  |-  ( A  e.  B  ->  E. x  x  =  A )
43pm4.71i 636 . . . 4  |-  ( A  e.  B  <->  ( A  e.  B  /\  E. x  x  =  A )
)
5 19.42v 1827 . . . 4  |-  ( E. x ( A  e.  B  /\  x  =  A )  <->  ( A  e.  B  /\  E. x  x  =  A )
)
6 eleq1 2495 . . . . . . 7  |-  ( A  =  x  ->  ( A  e.  B  <->  x  e.  B ) )
76eqcoms 2434 . . . . . 6  |-  ( x  =  A  ->  ( A  e.  B  <->  x  e.  B ) )
87pm5.32ri 642 . . . . 5  |-  ( ( A  e.  B  /\  x  =  A )  <->  ( x  e.  B  /\  x  =  A )
)
98exbii 1712 . . . 4  |-  ( E. x ( A  e.  B  /\  x  =  A )  <->  E. x
( x  e.  B  /\  x  =  A
) )
104, 5, 93bitr2i 276 . . 3  |-  ( A  e.  B  <->  E. x
( x  e.  B  /\  x  =  A
) )
11 vex 3083 . . . . . . 7  |-  x  e. 
_V
12 sneqbg 4170 . . . . . . 7  |-  ( x  e.  _V  ->  ( { x }  =  { A }  <->  x  =  A ) )
1311, 12ax-mp 5 . . . . . 6  |-  ( { x }  =  { A }  <->  x  =  A
)
14 eqcom 2431 . . . . . 6  |-  ( { x }  =  { A }  <->  { A }  =  { x } )
1513, 14bitr3i 254 . . . . 5  |-  ( x  =  A  <->  { A }  =  { x } )
1615anbi2i 698 . . . 4  |-  ( ( x  e.  B  /\  x  =  A )  <->  ( x  e.  B  /\  { A }  =  {
x } ) )
1716exbii 1712 . . 3  |-  ( E. x ( x  e.  B  /\  x  =  A )  <->  E. x
( x  e.  B  /\  { A }  =  { x } ) )
1810, 17bitri 252 . 2  |-  ( A  e.  B  <->  E. x
( x  e.  B  /\  { A }  =  { x } ) )
191, 2, 183bitr4ri 281 1  |-  ( A  e.  B  <->  { A }  e. sngl  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872   E.wrex 2772   _Vcvv 3080   {csn 3998  sngl bj-csngl 31527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-v 3082  df-dif 3439  df-un 3441  df-nul 3762  df-sn 3999  df-pr 4001  df-bj-sngl 31528
This theorem is referenced by:  bj-snglinv  31534  bj-tagci  31546  bj-tagcg  31547
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