Theorem bj-eltag 36943

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 ↔ (∃𝑥 ∈ 𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem bj-eltag

Step	Hyp	Ref	Expression
1		df-bj-tag 36941	. . 3 ⊢ tag 𝐵 = (sngl 𝐵 ∪ {∅})
2	1	eleq2i 2836	. 2 ⊢ (𝐴 ∈ tag 𝐵 ↔ 𝐴 ∈ (sngl 𝐵 ∪ {∅}))
3		elun 4176	. 2 ⊢ (𝐴 ∈ (sngl 𝐵 ∪ {∅}) ↔ (𝐴 ∈ sngl 𝐵 ∨ 𝐴 ∈ {∅}))
4		bj-elsngl 36934	. . 3 ⊢ (𝐴 ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 = {𝑥})
5		0ex 5325	. . . 4 ⊢ ∅ ∈ V
6	5	elsn2 4687	. . 3 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
7	4, 6	orbi12i 913	. 2 ⊢ ((𝐴 ∈ sngl 𝐵 ∨ 𝐴 ∈ {∅}) ↔ (∃𝑥 ∈ 𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅))
8	2, 3, 7	3bitri 297	1 ⊢ (𝐴 ∈ tag 𝐵 ↔ (∃𝑥 ∈ 𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅))

Syntax hints:   ↔ wb 206   ∨ wo 846   = wceq 1537   ∈ wcel 2108  ∃wrex 3076   ∪ cun 3974  ∅c0 4352  {csn 4648  sngl bj-csngl 36931  tag bj-ctag 36940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rex 3077  df-v 3490  df-dif 3979  df-un 3981  df-nul 4353  df-sn 4649  df-pr 4651  df-bj-sngl 36932  df-bj-tag 36941

This theorem is referenced by: (None)