Theorem bj-sngltag 37567

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 37566	. 2 ⊢ ({𝐴} ∈ sngl 𝐵 → {𝐴} ∈ tag 𝐵)
2		df-bj-tag 37559	. . . 4 ⊢ tag 𝐵 = (sngl 𝐵 ∪ {∅})
3	2	eleq2i 2862	. . 3 ⊢ ({𝐴} ∈ tag 𝐵 ↔ {𝐴} ∈ (sngl 𝐵 ∪ {∅}))
4		elun 4115	. . . 4 ⊢ ({𝐴} ∈ (sngl 𝐵 ∪ {∅}) ↔ ({𝐴} ∈ sngl 𝐵 ∨ {𝐴} ∈ {∅}))
5		idd 25	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ sngl 𝐵 → {𝐴} ∈ sngl 𝐵))
6		elsni 4611	. . . . . 6 ⊢ ({𝐴} ∈ {∅} → {𝐴} = ∅)
7		snprc 4688	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
8		elex 3483	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
9	8	pm2.24d 152	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ V → {𝐴} ∈ sngl 𝐵))
10	7, 9	biimtrrid 246	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = ∅ → {𝐴} ∈ sngl 𝐵))
11	6, 10	syl5 35	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ {∅} → {𝐴} ∈ sngl 𝐵))
12	5, 11	jaod 872	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (({𝐴} ∈ sngl 𝐵 ∨ {𝐴} ∈ {∅}) → {𝐴} ∈ sngl 𝐵))
13	4, 12	biimtrid 245	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ (sngl 𝐵 ∪ {∅}) → {𝐴} ∈ sngl 𝐵))
14	3, 13	biimtrid 245	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ tag 𝐵 → {𝐴} ∈ sngl 𝐵))
15	1, 14	impbid2 229	1 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ sngl 𝐵 ↔ {𝐴} ∈ tag 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 860   = wceq 1568   ∈ wcel 2150  Vcvv 3462   ∪ cun 3911  ∅c0 4294  {csn 4594  sngl bj-csngl 37549  tag bj-ctag 37558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-v 3464  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-sn 4595  df-bj-tag 37559

This theorem is referenced by:  bj-tagcg  37569  bj-taginv  37570