Theorem epelg 5487

Description: The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5489 and closed form of epeli 5488. (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 5071	. . . 4 ⊢ (𝐴 E 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ E )
2		0nelopab 5471	. . . . . . . 8 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
3		df-eprel 5486	. . . . . . . . . 10 ⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
4	3	eqcomi 2747	. . . . . . . . 9 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} = E
5	4	eleq2i 2830	. . . . . . . 8 ⊢ (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} ↔ ∅ ∈ E )
6	2, 5	mtbi 321	. . . . . . 7 ⊢ ¬ ∅ ∈ E
7		eleq1 2826	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ∅ → (⟨𝐴, 𝐵⟩ ∈ E ↔ ∅ ∈ E ))
8	6, 7	mtbiri 326	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ = ∅ → ¬ ⟨𝐴, 𝐵⟩ ∈ E )
9	8	con2i 139	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ E → ¬ ⟨𝐴, 𝐵⟩ = ∅)
10		opprc1 4825	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
11	9, 10	nsyl2 141	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ E → 𝐴 ∈ V)
12	1, 11	sylbi 216	. . 3 ⊢ (𝐴 E 𝐵 → 𝐴 ∈ V)
13	12	a1i 11	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 → 𝐴 ∈ V))
14		elex 3440	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
15	14	a1i 11	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 → 𝐴 ∈ V))
16		eleq12 2828	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵))
17	16, 3	brabga 5440	. . 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 380	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ∅c0 4253  ⟨cop 4564   class class class wbr 5070  {copab 5132   E cep 5485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-eprel 5486

This theorem is referenced by:  epeli  5488  efrirr  5561  efrn2lp  5562  epin  5992  predep  6222  epne3  7601  cnfcomlem  9387  fpwwe2lem5  10322  ltpiord  10574  orvcelval  32335  bj-epelb  35167  brcnvep  36331  epelon2  41026  alephiso2  41054