Theorem epelg 5542

Description: The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5544 and closed form of epeli 5543. Definition 1.6 of [Schloeder] p. 1. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) (Proof shortened by BJ, 14-Jul-2023.)

Assertion

Ref	Expression
epelg	⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))

Proof of Theorem epelg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-br 5111	. . . 4 ⊢ (𝐴 E 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ E )
2		0nelopab 5530	. . . . . . . 8 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
3		df-eprel 5541	. . . . . . . . . 10 ⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
4	3	eqcomi 2739	. . . . . . . . 9 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} = E
5	4	eleq2i 2821	. . . . . . . 8 ⊢ (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} ↔ ∅ ∈ E )
6	2, 5	mtbi 322	. . . . . . 7 ⊢ ¬ ∅ ∈ E
7		eleq1 2817	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ∅ → (⟨𝐴, 𝐵⟩ ∈ E ↔ ∅ ∈ E ))
8	6, 7	mtbiri 327	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ = ∅ → ¬ ⟨𝐴, 𝐵⟩ ∈ E )
9	8	con2i 139	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ E → ¬ ⟨𝐴, 𝐵⟩ = ∅)
10		opprc1 4864	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
11	9, 10	nsyl2 141	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ E → 𝐴 ∈ V)
12	1, 11	sylbi 217	. . 3 ⊢ (𝐴 E 𝐵 → 𝐴 ∈ V)
13	12	a1i 11	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 → 𝐴 ∈ V))
14		elex 3471	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
15	14	a1i 11	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 → 𝐴 ∈ V))
16		eleq12 2819	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵))
17	16, 3	brabga 5497	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
18	17	expcom 413	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)))
19	13, 15, 18	pm5.21ndd 379	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ∅c0 4299  ⟨cop 4598   class class class wbr 5110  {copab 5172   E cep 5540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-eprel 5541

This theorem is referenced by:  epeli  5543  efrirr  5621  efrn2lp  5622  epin  6069  predep  6306  epne3  7752  cnfcomlem  9659  fpwwe2lem5  10595  ltpiord  10847  orvcelval  34467  bj-epelb  37064  brcnvep  38261  onsupuni  43225  oninfint  43232  onepsuc  43248  cantnfresb  43320  epelon2  43517  alephiso2  43554