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Theorem bj-eltag 32949
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 32947 . . 3 tag 𝐵 = (sngl 𝐵 ∪ {∅})
21eleq2i 2692 . 2 (𝐴 ∈ tag 𝐵𝐴 ∈ (sngl 𝐵 ∪ {∅}))
3 elun 3751 . 2 (𝐴 ∈ (sngl 𝐵 ∪ {∅}) ↔ (𝐴 ∈ sngl 𝐵𝐴 ∈ {∅}))
4 bj-elsngl 32940 . . 3 (𝐴 ∈ sngl 𝐵 ↔ ∃𝑥𝐵 𝐴 = {𝑥})
5 0ex 4788 . . . 4 ∅ ∈ V
65elsn2 4209 . . 3 (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
74, 6orbi12i 543 . 2 ((𝐴 ∈ sngl 𝐵𝐴 ∈ {∅}) ↔ (∃𝑥𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅))
82, 3, 73bitri 286 1 (𝐴 ∈ tag 𝐵 ↔ (∃𝑥𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 383   = wceq 1482  wcel 1989  wrex 2912  cun 3570  c0 3913  {csn 4175  sngl bj-csngl 32937  tag bj-ctag 32946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-v 3200  df-dif 3575  df-un 3577  df-nul 3914  df-sn 4176  df-pr 4178  df-bj-sngl 32938  df-bj-tag 32947
This theorem is referenced by: (None)
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