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Theorem bj-snglc 31633
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 2762 . 2  |-  ( E. x  e.  B  { A }  =  {
x }  <->  E. x
( x  e.  B  /\  { A }  =  { x } ) )
2 bj-elsngl 31632 . 2  |-  ( { A }  e. sngl  B  <->  E. x  e.  B  { A }  =  {
x } )
3 elisset 3043 . . . . 5  |-  ( A  e.  B  ->  E. x  x  =  A )
43pm4.71i 644 . . . 4  |-  ( A  e.  B  <->  ( A  e.  B  /\  E. x  x  =  A )
)
5 19.42v 1842 . . . 4  |-  ( E. x ( A  e.  B  /\  x  =  A )  <->  ( A  e.  B  /\  E. x  x  =  A )
)
6 eleq1 2537 . . . . . . 7  |-  ( A  =  x  ->  ( A  e.  B  <->  x  e.  B ) )
76eqcoms 2479 . . . . . 6  |-  ( x  =  A  ->  ( A  e.  B  <->  x  e.  B ) )
87pm5.32ri 650 . . . . 5  |-  ( ( A  e.  B  /\  x  =  A )  <->  ( x  e.  B  /\  x  =  A )
)
98exbii 1726 . . . 4  |-  ( E. x ( A  e.  B  /\  x  =  A )  <->  E. x
( x  e.  B  /\  x  =  A
) )
104, 5, 93bitr2i 281 . . 3  |-  ( A  e.  B  <->  E. x
( x  e.  B  /\  x  =  A
) )
11 vex 3034 . . . . . . 7  |-  x  e. 
_V
12 sneqbg 4134 . . . . . . 7  |-  ( x  e.  _V  ->  ( { x }  =  { A }  <->  x  =  A ) )
1311, 12ax-mp 5 . . . . . 6  |-  ( { x }  =  { A }  <->  x  =  A
)
14 eqcom 2478 . . . . . 6  |-  ( { x }  =  { A }  <->  { A }  =  { x } )
1513, 14bitr3i 259 . . . . 5  |-  ( x  =  A  <->  { A }  =  { x } )
1615anbi2i 708 . . . 4  |-  ( ( x  e.  B  /\  x  =  A )  <->  ( x  e.  B  /\  { A }  =  {
x } ) )
1716exbii 1726 . . 3  |-  ( E. x ( x  e.  B  /\  x  =  A )  <->  E. x
( x  e.  B  /\  { A }  =  { x } ) )
1810, 17bitri 257 . 2  |-  ( A  e.  B  <->  E. x
( x  e.  B  /\  { A }  =  { x } ) )
191, 2, 183bitr4ri 286 1  |-  ( A  e.  B  <->  { A }  e. sngl  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   E.wrex 2757   _Vcvv 3031   {csn 3959  sngl bj-csngl 31629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-v 3033  df-dif 3393  df-un 3395  df-nul 3723  df-sn 3960  df-pr 3962  df-bj-sngl 31630
This theorem is referenced by:  bj-snglinv  31636  bj-tagci  31648  bj-tagcg  31649
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