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Theorem bj-eltag 36978
Description: Characterization of the elements of the tagging of a class. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-eltag (𝐴 ∈ tag 𝐵 ↔ (∃𝑥𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem bj-eltag
StepHypRef Expression
1 df-bj-tag 36976 . . 3 tag 𝐵 = (sngl 𝐵 ∪ {∅})
21eleq2i 2833 . 2 (𝐴 ∈ tag 𝐵𝐴 ∈ (sngl 𝐵 ∪ {∅}))
3 elun 4153 . 2 (𝐴 ∈ (sngl 𝐵 ∪ {∅}) ↔ (𝐴 ∈ sngl 𝐵𝐴 ∈ {∅}))
4 bj-elsngl 36969 . . 3 (𝐴 ∈ sngl 𝐵 ↔ ∃𝑥𝐵 𝐴 = {𝑥})
5 0ex 5307 . . . 4 ∅ ∈ V
65elsn2 4665 . . 3 (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
74, 6orbi12i 915 . 2 ((𝐴 ∈ sngl 𝐵𝐴 ∈ {∅}) ↔ (∃𝑥𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅))
82, 3, 73bitri 297 1 (𝐴 ∈ tag 𝐵 ↔ (∃𝑥𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wo 848   = wceq 1540  wcel 2108  wrex 3070  cun 3949  c0 4333  {csn 4626  sngl bj-csngl 36966  tag bj-ctag 36975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3071  df-v 3482  df-dif 3954  df-un 3956  df-nul 4334  df-sn 4627  df-pr 4629  df-bj-sngl 36967  df-bj-tag 36976
This theorem is referenced by: (None)
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