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Theorem bj-snglc 33608
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 2820 . 2  |-  ( E. x  e.  B  { A }  =  {
x }  <->  E. x
( x  e.  B  /\  { A }  =  { x } ) )
2 bj-elsngl 33607 . 2  |-  ( { A }  e. sngl  B  <->  E. x  e.  B  { A }  =  {
x } )
3 elisset 3124 . . . . 5  |-  ( A  e.  B  ->  E. x  x  =  A )
43pm4.71i 632 . . . 4  |-  ( A  e.  B  <->  ( A  e.  B  /\  E. x  x  =  A )
)
5 19.42v 1949 . . . 4  |-  ( E. x ( A  e.  B  /\  x  =  A )  <->  ( A  e.  B  /\  E. x  x  =  A )
)
6 eleq1 2539 . . . . . . 7  |-  ( A  =  x  ->  ( A  e.  B  <->  x  e.  B ) )
76eqcoms 2479 . . . . . 6  |-  ( x  =  A  ->  ( A  e.  B  <->  x  e.  B ) )
87pm5.32ri 638 . . . . 5  |-  ( ( A  e.  B  /\  x  =  A )  <->  ( x  e.  B  /\  x  =  A )
)
98exbii 1644 . . . 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 3116 . . . . . . 7  |-  x  e. 
_V
12 sneqbg 4197 . . . . . . 7  |-  ( x  e.  _V  ->  ( { x }  =  { A }  <->  x  =  A ) )
1311, 12ax-mp 5 . . . . . 6  |-  ( { x }  =  { A }  <->  x  =  A
)
14 eqcom 2476 . . . . . 6  |-  ( { x }  =  { A }  <->  { A }  =  { x } )
1513, 14bitr3i 251 . . . . 5  |-  ( x  =  A  <->  { A }  =  { x } )
1615anbi2i 694 . . . 4  |-  ( ( x  e.  B  /\  x  =  A )  <->  ( x  e.  B  /\  { A }  =  {
x } ) )
1716exbii 1644 . . 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 369    = wceq 1379   E.wex 1596    e. wcel 1767   E.wrex 2815   _Vcvv 3113   {csn 4027  sngl bj-csngl 33604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-v 3115  df-dif 3479  df-un 3481  df-nul 3786  df-sn 4028  df-pr 4030  df-bj-sngl 33605
This theorem is referenced by:  bj-snglinv  33611  bj-tagci  33623  bj-tagcg  33624
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