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Theorem bj-taginv 34704
 Description: Inverse of tagging. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-taginv 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴}
Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-taginv
StepHypRef Expression
1 bj-snglinv 34690 . 2 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}
2 bj-sngltag 34701 . . . 4 (𝑥 ∈ V → ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴))
32elv 3416 . . 3 ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴)
43abbii 2824 . 2 {𝑥 ∣ {𝑥} ∈ sngl 𝐴} = {𝑥 ∣ {𝑥} ∈ tag 𝐴}
51, 4eqtri 2782 1 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1539   ∈ wcel 2112  {cab 2736  Vcvv 3410  {csn 4523  sngl bj-csngl 34683  tag bj-ctag 34692 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-rex 3077  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-sn 4524  df-pr 4526  df-bj-sngl 34684  df-bj-tag 34693 This theorem is referenced by:  bj-projval  34714
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