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Theorem bj-eltag 36959
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 36957 . . 3 tag 𝐵 = (sngl 𝐵 ∪ {∅})
21eleq2i 2830 . 2 (𝐴 ∈ tag 𝐵𝐴 ∈ (sngl 𝐵 ∪ {∅}))
3 elun 4162 . 2 (𝐴 ∈ (sngl 𝐵 ∪ {∅}) ↔ (𝐴 ∈ sngl 𝐵𝐴 ∈ {∅}))
4 bj-elsngl 36950 . . 3 (𝐴 ∈ sngl 𝐵 ↔ ∃𝑥𝐵 𝐴 = {𝑥})
5 0ex 5312 . . . 4 ∅ ∈ V
65elsn2 4669 . . 3 (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
74, 6orbi12i 914 . 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 847   = wceq 1536  wcel 2105  wrex 3067  cun 3960  c0 4338  {csn 4630  sngl bj-csngl 36947  tag bj-ctag 36956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rex 3068  df-v 3479  df-dif 3965  df-un 3967  df-nul 4339  df-sn 4631  df-pr 4633  df-bj-sngl 36948  df-bj-tag 36957
This theorem is referenced by: (None)
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