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Theorem bj-taginv 34945
Description: Inverse of tagging. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-taginv  |-  A  =  { x  |  {
x }  e. tag  A }
Distinct variable group:    x, A

Proof of Theorem bj-taginv
StepHypRef Expression
1 bj-snglinv 34931 . 2  |-  A  =  { x  |  {
x }  e. sngl  A }
2 vex 3109 . . . 4  |-  x  e. 
_V
3 bj-sngltag 34942 . . . 4  |-  ( x  e.  _V  ->  ( { x }  e. sngl  A  <->  { x }  e. tag  A ) )
42, 3ax-mp 5 . . 3  |-  ( { x }  e. sngl  A  <->  { x }  e. tag  A
)
54abbii 2588 . 2  |-  { x  |  { x }  e. sngl  A }  =  { x  |  { x }  e. tag  A }
61, 5eqtri 2483 1  |-  A  =  { x  |  {
x }  e. tag  A }
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1398    e. wcel 1823   {cab 2439   _Vcvv 3106   {csn 4016  sngl bj-csngl 34924  tag bj-ctag 34933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-sn 4017  df-pr 4019  df-bj-sngl 34925  df-bj-tag 34934
This theorem is referenced by:  bj-projval  34955
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