Theorem bj-eltag 37812

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 37810	. . 3 ⊢ tag 𝐵 = (sngl 𝐵 ∪ {∅})
2	1	eleq2i 2852	. 2 ⊢ (𝐴 ∈ tag 𝐵 ↔ 𝐴 ∈ (sngl 𝐵 ∪ {∅}))
3		elun 4099	. 2 ⊢ (𝐴 ∈ (sngl 𝐵 ∪ {∅}) ↔ (𝐴 ∈ sngl 𝐵 ∨ 𝐴 ∈ {∅}))
4		bj-elsngl 37803	. . 3 ⊢ (𝐴 ∈ sngl 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 = {𝑥})
5		0ex 5260	. . . 4 ⊢ ∅ ∈ V
6	5	elsn2 4625	. . 3 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
7	4, 6	orbi12i 928	. 2 ⊢ ((𝐴 ∈ sngl 𝐵 ∨ 𝐴 ∈ {∅}) ↔ (∃𝑥 ∈ 𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅))
8	2, 3, 7	3bitri 300	1 ⊢ (𝐴 ∈ tag 𝐵 ↔ (∃𝑥 ∈ 𝐵 𝐴 = {𝑥} ∨ 𝐴 = ∅))

This proof depends on syntax axioms:   ↔ wb 209   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∃wrex 3086   ∪ cun 3896  ∅c0 4278  {csn 4583  sngl bj-csngl 37800  tag bj-ctag 37809

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rex 3087  df-v 3452  df-dif 3901  df-un 3903  df-nul 4279  df-sn 4584  df-pr 4586  df-bj-sngl 37801  df-bj-tag 37810

This theorem is used by: (None)