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Theorem bj-taginv 32812
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 32798 . 2  |-  A  =  { x  |  {
x }  e. sngl  A }
2 vex 3081 . . . 4  |-  x  e. 
_V
3 bj-sngltag 32809 . . . 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 1370    e. wcel 1758   {cab 2439   _Vcvv 3078   {csn 3986  sngl bj-csngl 32791  tag bj-ctag 32800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rex 2805  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-sn 3987  df-pr 3989  df-bj-sngl 32792  df-bj-tag 32801
This theorem is referenced by:  bj-projval  32822
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