Theorem bj-epelg 35949

Description: The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5584 and closed form of epeli 5583. (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 5733 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 5828	. . . 4 ⊢ Rel E
2	1	brrelex1i 5733	. . 3 ⊢ (𝐴 E 𝐵 → 𝐴 ∈ V)
3	2	a1i 11	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 → 𝐴 ∈ V))
4		elex 3493	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
5	4	a1i 11	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 → 𝐴 ∈ V))
6		eleq12 2824	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵))
7		df-eprel 5581	. . . 4 ⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
8	6, 7	brabga 5535	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
9	8	expcom 415	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)))
10	3, 5, 9	pm5.21ndd 381	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2107  Vcvv 3475   class class class wbr 5149   E cep 5580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-eprel 5581  df-xp 5683  df-rel 5684

This theorem is referenced by: (None)