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Theorem bj-eltag 35380
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 35378 . . 3 tag 𝐵 = (sngl 𝐵 ∪ {∅})
21eleq2i 2830 . 2 (𝐴 ∈ tag 𝐵𝐴 ∈ (sngl 𝐵 ∪ {∅}))
3 elun 4107 . 2 (𝐴 ∈ (sngl 𝐵 ∪ {∅}) ↔ (𝐴 ∈ sngl 𝐵𝐴 ∈ {∅}))
4 bj-elsngl 35371 . . 3 (𝐴 ∈ sngl 𝐵 ↔ ∃𝑥𝐵 𝐴 = {𝑥})
5 0ex 5263 . . . 4 ∅ ∈ V
65elsn2 4624 . . 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 205  wo 846   = wceq 1542  wcel 2107  wrex 3072  cun 3907  c0 4281  {csn 4585  sngl bj-csngl 35368  tag bj-ctag 35377
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-rex 3073  df-v 3446  df-dif 3912  df-un 3914  df-nul 4282  df-sn 4586  df-pr 4588  df-bj-sngl 35369  df-bj-tag 35378
This theorem is referenced by: (None)
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