Theorem epelg 5564

Description: The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5566 and closed form of epeli 5565. 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 5552	. . . . . . . 8 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
3		df-eprel 5563	. . . . . . . . . 10 ⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
4	3	eqcomi 2772	. . . . . . . . 9 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} = E
5	4	eleq2i 2855	. . . . . . . 8 ⊢ (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} ↔ ∅ ∈ E )
6	2, 5	mtbi 325	. . . . . . 7 ⊢ ¬ ∅ ∈ E
7		eleq1 2851	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ∅ → (⟨𝐴, 𝐵⟩ ∈ E ↔ ∅ ∈ E ))
8	6, 7	mtbiri 330	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ = ∅ → ¬ ⟨𝐴, 𝐵⟩ ∈ E )
9	8	con2i 140	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ E → ¬ ⟨𝐴, 𝐵⟩ = ∅)
10		opprc1 4863	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
11	9, 10	nsyl2 142	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ E → 𝐴 ∈ V)
12	1, 11	sylbi 220	. . 3 ⊢ (𝐴 E 𝐵 → 𝐴 ∈ V)
13	12	a1i 11	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 → 𝐴 ∈ V))
14		elex 3476	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
15	14	a1i 11	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 → 𝐴 ∈ V))
16		eleq12 2853	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵))
17	16, 3	brabga 5520	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
18	17	expcom 418	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)))
19	13, 15, 18	pm5.21ndd 382	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ∅c0 4287  ⟨cop 4596   class class class wbr 5110  {copab 5174   E cep 5562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-eprel 5563

This theorem is referenced by:  epeli  5565  efrirr  5643  efrn2lp  5644  epin  6099  predep  6333  epne3  7773  cnfcomlem  9669  fpwwe2lem5  10621  ltpiord  10873  tgelrnpln  29039  orvcelval  34840  bj-epelb  37686  brcnvep  38900  onsupuni  43939  oninfint  43946  onepsuc  43962  cantnfresb  44034  epelon2  44230  alephiso2  44267