Theorem bj-epelg 37670

Description: The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5564 and closed form of epeli 5563. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) TODO: move it to the main section after reordering to have brrelex1i 5717 available. (Proof shortened by BJ, 14-Jul-2023.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem bj-epelg

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

Step	Hyp	Ref	Expression
1		rele 5814	. . . 4 ⊢ Rel E
2	1	brrelex1i 5717	. . 3 ⊢ (𝐴 E 𝐵 → 𝐴 ∈ V)
3	2	a1i 11	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 → 𝐴 ∈ V))
4		elex 3474	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
5	4	a1i 11	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 → 𝐴 ∈ V))
6		eleq12 2851	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵))
7		df-eprel 5561	. . . 4 ⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
8	6, 7	brabga 5518	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
9	8	expcom 418	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)))
10	3, 5, 9	pm5.21ndd 382	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2141  Vcvv 3453   class class class wbr 5108   E cep 5560

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 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-eprel 5561  df-xp 5667  df-rel 5668

This theorem is referenced by: (None)