Theorem bj-eltag 34291

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 34289	. . 3 ⊢ tag 𝐵 = (sngl 𝐵 ∪ {∅})
2	1	eleq2i 2906	. 2 ⊢ (𝐴 ∈ tag 𝐵 ↔ 𝐴 ∈ (sngl 𝐵 ∪ {∅}))
3		elun 4127	. 2 ⊢ (𝐴 ∈ (sngl 𝐵 ∪ {∅}) ↔ (𝐴 ∈ sngl 𝐵 ∨ 𝐴 ∈ {∅}))
4		bj-elsngl 34282	. . 3 ⊢ (𝐴 ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 = {𝑥})
5		0ex 5213	. . . 4 ⊢ ∅ ∈ V
6	5	elsn2 4606	. . 3 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
7	4, 6	orbi12i 911	. 2 ⊢ ((𝐴 ∈ sngl 𝐵 ∨ 𝐴 ∈ {∅}) ↔ (∃𝑥 ∈ 𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅))
8	2, 3, 7	3bitri 299	1 ⊢ (𝐴 ∈ tag 𝐵 ↔ (∃𝑥 ∈ 𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅))

Syntax hints:   ↔ wb 208   ∨ wo 843   = wceq 1537   ∈ wcel 2114  ∃wrex 3141   ∪ cun 3936  ∅c0 4293  {csn 4569  sngl bj-csngl 34279  tag bj-ctag 34288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rex 3146  df-v 3498  df-dif 3941  df-un 3943  df-nul 4294  df-sn 4570  df-pr 4572  df-bj-sngl 34280  df-bj-tag 34289

This theorem is referenced by: (None)