Theorem epelg 5522

Description: The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5524 and closed form of epeli 5523. 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 5096	. . . 4 ⊢ (𝐴 E 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ E )
2		0nelopab 5510	. . . . . . . 8 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
3		df-eprel 5521	. . . . . . . . . 10 ⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
4	3	eqcomi 2742	. . . . . . . . 9 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} = E
5	4	eleq2i 2825	. . . . . . . 8 ⊢ (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} ↔ ∅ ∈ E )
6	2, 5	mtbi 322	. . . . . . 7 ⊢ ¬ ∅ ∈ E
7		eleq1 2821	. . . . . . 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 4850	. . . . 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 3458	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
15	14	a1i 11	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 → 𝐴 ∈ V))
16		eleq12 2823	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵))
17	16, 3	brabga 5479	. . 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 1541   ∈ wcel 2113  Vcvv 3437  ∅c0 4282  ⟨cop 4583   class class class wbr 5095  {copab 5157   E cep 5520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-eprel 5521

This theorem is referenced by:  epeli  5523  efrirr  5601  efrn2lp  5602  epin  6051  predep  6285  epne3  7715  cnfcomlem  9600  fpwwe2lem5  10537  ltpiord  10789  orvcelval  34554  bj-epelb  37186  brcnvep  38375  onsupuni  43386  oninfint  43393  onepsuc  43409  cantnfresb  43481  epelon2  43678  alephiso2  43715