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Theorem bj-elsngl 34669
Description: Characterization of the elements of the singletonization of a class. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-elsngl  |-  ( A  e. sngl  B  <->  E. x  e.  B  A  =  { x } )
Distinct variable groups:    x, A    x, B

Proof of Theorem bj-elsngl
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-clel 2452 . 2  |-  ( A  e. sngl  B  <->  E. y
( y  =  A  /\  y  e. sngl  B
) )
2 df-bj-sngl 34667 . . . . 5  |- sngl  B  =  { y  |  E. x  e.  B  y  =  { x } }
32abeq2i 2584 . . . 4  |-  ( y  e. sngl  B  <->  E. x  e.  B  y  =  { x } )
43anbi2i 694 . . 3  |-  ( ( y  =  A  /\  y  e. sngl  B )  <->  ( y  =  A  /\  E. x  e.  B  y  =  { x }
) )
54exbii 1668 . 2  |-  ( E. y ( y  =  A  /\  y  e. sngl  B )  <->  E. y
( y  =  A  /\  E. x  e.  B  y  =  {
x } ) )
6 r19.42v 3012 . . . . 5  |-  ( E. x  e.  B  ( y  =  A  /\  y  =  { x } )  <->  ( y  =  A  /\  E. x  e.  B  y  =  { x } ) )
76bicomi 202 . . . 4  |-  ( ( y  =  A  /\  E. x  e.  B  y  =  { x }
)  <->  E. x  e.  B  ( y  =  A  /\  y  =  {
x } ) )
87exbii 1668 . . 3  |-  ( E. y ( y  =  A  /\  E. x  e.  B  y  =  { x } )  <->  E. y E. x  e.  B  ( y  =  A  /\  y  =  { x } ) )
9 rexcom4 3129 . . . 4  |-  ( E. x  e.  B  E. y ( y  =  A  /\  y  =  { x } )  <->  E. y E. x  e.  B  ( y  =  A  /\  y  =  { x } ) )
109bicomi 202 . . 3  |-  ( E. y E. x  e.  B  ( y  =  A  /\  y  =  { x } )  <->  E. x  e.  B  E. y ( y  =  A  /\  y  =  { x } ) )
11 eqcom 2466 . . . . . 6  |-  ( A  =  { x }  <->  { x }  =  A )
12 snex 4697 . . . . . . 7  |-  { x }  e.  _V
1312eqvinc 3226 . . . . . 6  |-  ( { x }  =  A  <->  E. y ( y  =  { x }  /\  y  =  A )
)
14 exancom 1672 . . . . . 6  |-  ( E. y ( y  =  { x }  /\  y  =  A )  <->  E. y ( y  =  A  /\  y  =  { x } ) )
1511, 13, 143bitri 271 . . . . 5  |-  ( A  =  { x }  <->  E. y ( y  =  A  /\  y  =  { x } ) )
1615bicomi 202 . . . 4  |-  ( E. y ( y  =  A  /\  y  =  { x } )  <-> 
A  =  { x } )
1716rexbii 2959 . . 3  |-  ( E. x  e.  B  E. y ( y  =  A  /\  y  =  { x } )  <->  E. x  e.  B  A  =  { x } )
188, 10, 173bitri 271 . 2  |-  ( E. y ( y  =  A  /\  E. x  e.  B  y  =  { x } )  <->  E. x  e.  B  A  =  { x } )
191, 5, 183bitri 271 1  |-  ( A  e. sngl  B  <->  E. x  e.  B  A  =  { x } )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395   E.wex 1613    e. wcel 1819   E.wrex 2808   {csn 4032  sngl bj-csngl 34666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3474  df-un 3476  df-nul 3794  df-sn 4033  df-pr 4035  df-bj-sngl 34667
This theorem is referenced by:  bj-snglc  34670  bj-snglss  34671  bj-0nelsngl  34672  bj-eltag  34678
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