Theorem bj-sngltag 33463

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 33462	. 2 ⊢ ({𝐴} ∈ sngl 𝐵 → {𝐴} ∈ tag 𝐵)
2		df-bj-tag 33455	. . . 4 ⊢ tag 𝐵 = (sngl 𝐵 ∪ {∅})
3	2	eleq2i 2870	. . 3 ⊢ ({𝐴} ∈ tag 𝐵 ↔ {𝐴} ∈ (sngl 𝐵 ∪ {∅}))
4		elun 3951	. . . 4 ⊢ ({𝐴} ∈ (sngl 𝐵 ∪ {∅}) ↔ ({𝐴} ∈ sngl 𝐵 ∨ {𝐴} ∈ {∅}))
5		idd 24	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ sngl 𝐵 → {𝐴} ∈ sngl 𝐵))
6		elsni 4385	. . . . . 6 ⊢ ({𝐴} ∈ {∅} → {𝐴} = ∅)
7		snprc 4442	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
8		elex 3400	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
9	8	pm2.24d 149	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ V → {𝐴} ∈ sngl 𝐵))
10	7, 9	syl5bir 235	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = ∅ → {𝐴} ∈ sngl 𝐵))
11	6, 10	syl5 34	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ {∅} → {𝐴} ∈ sngl 𝐵))
12	5, 11	jaod 886	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (({𝐴} ∈ sngl 𝐵 ∨ {𝐴} ∈ {∅}) → {𝐴} ∈ sngl 𝐵))
13	4, 12	syl5bi 234	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ (sngl 𝐵 ∪ {∅}) → {𝐴} ∈ sngl 𝐵))
14	3, 13	syl5bi 234	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ tag 𝐵 → {𝐴} ∈ sngl 𝐵))
15	1, 14	impbid2 218	1 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ sngl 𝐵 ↔ {𝐴} ∈ tag 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∨ wo 874   = wceq 1653   ∈ wcel 2157  Vcvv 3385   ∪ cun 3767  ∅c0 4115  {csn 4368  sngl bj-csngl 33445  tag bj-ctag 33454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-sn 4369  df-bj-tag 33455

This theorem is referenced by:  bj-tagcg  33465  bj-taginv  33466