Theorem bj-sngltag 36519

Description: The singletonization and the tagging of a set contain the same singletons. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-sngltag	⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ sngl 𝐵 ↔ {𝐴} ∈ tag 𝐵))

Proof of Theorem bj-sngltag

Step	Hyp	Ref	Expression
1		bj-sngltagi 36518	. 2 ⊢ ({𝐴} ∈ sngl 𝐵 → {𝐴} ∈ tag 𝐵)
2		df-bj-tag 36511	. . . 4 ⊢ tag 𝐵 = (sngl 𝐵 ∪ {∅})
3	2	eleq2i 2817	. . 3 ⊢ ({𝐴} ∈ tag 𝐵 ↔ {𝐴} ∈ (sngl 𝐵 ∪ {∅}))
4		elun 4141	. . . 4 ⊢ ({𝐴} ∈ (sngl 𝐵 ∪ {∅}) ↔ ({𝐴} ∈ sngl 𝐵 ∨ {𝐴} ∈ {∅}))
5		idd 24	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ sngl 𝐵 → {𝐴} ∈ sngl 𝐵))
6		elsni 4641	. . . . . 6 ⊢ ({𝐴} ∈ {∅} → {𝐴} = ∅)
7		snprc 4717	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
8		elex 3482	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
9	8	pm2.24d 151	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ V → {𝐴} ∈ sngl 𝐵))
10	7, 9	biimtrrid 242	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = ∅ → {𝐴} ∈ sngl 𝐵))
11	6, 10	syl5 34	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ {∅} → {𝐴} ∈ sngl 𝐵))
12	5, 11	jaod 857	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (({𝐴} ∈ sngl 𝐵 ∨ {𝐴} ∈ {∅}) → {𝐴} ∈ sngl 𝐵))
13	4, 12	biimtrid 241	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ (sngl 𝐵 ∪ {∅}) → {𝐴} ∈ sngl 𝐵))
14	3, 13	biimtrid 241	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ tag 𝐵 → {𝐴} ∈ sngl 𝐵))
15	1, 14	impbid2 225	1 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ sngl 𝐵 ↔ {𝐴} ∈ tag 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∨ wo 845   = wceq 1533   ∈ wcel 2098  Vcvv 3463   ∪ cun 3937  ∅c0 4318  {csn 4624  sngl bj-csngl 36501  tag bj-ctag 36510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-sn 4625  df-bj-tag 36511

This theorem is referenced by:  bj-tagcg  36521  bj-taginv  36522