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 ↔ (∃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 36976	. . 3 ⊢ tag 𝐵 = (sngl 𝐵 ∪ {∅})
2	1	eleq2i 2833	. 2 ⊢ (𝐴 ∈ tag 𝐵 ↔ 𝐴 ∈ (sngl 𝐵 ∪ {∅}))
3		elun 4153	. 2 ⊢ (𝐴 ∈ (sngl 𝐵 ∪ {∅}) ↔ (𝐴 ∈ sngl 𝐵 ∨ 𝐴 ∈ {∅}))
4		bj-elsngl 36969	. . 3 ⊢ (𝐴 ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 = {𝑥})
5		0ex 5307	. . . 4 ⊢ ∅ ∈ V
6	5	elsn2 4665	. . 3 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
7	4, 6	orbi12i 915	. 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 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)