Theorem bj-epelb 37050

Description: Two classes are related by the membership relation if and only if they are related by the membership relation (i.e., the first is an element of the second) and the second is a set (hence so is the first). TODO: move to Main after reordering to have brrelex2i 5688 available. Check if it is shorter to prove bj-epelg 37049 first or bj-epelb 37050 first. (Contributed by BJ, 14-Jul-2023.)

Assertion

Ref	Expression
bj-epelb	⊢ (𝐴 E 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ V))

Proof of Theorem bj-epelb

Step	Hyp	Ref	Expression
1		rele 5781	. . . 4 ⊢ Rel E
2	1	brrelex2i 5688	. . 3 ⊢ (𝐴 E 𝐵 → 𝐵 ∈ V)
3	2	pm4.71i 559	. 2 ⊢ (𝐴 E 𝐵 ↔ (𝐴 E 𝐵 ∧ 𝐵 ∈ V))
4		epelg 5532	. . 3 ⊢ (𝐵 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
5	4	pm5.32ri 575	. 2 ⊢ ((𝐴 E 𝐵 ∧ 𝐵 ∈ V) ↔ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ V))
6	3, 5	bitri 275	1 ⊢ (𝐴 E 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ V))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  Vcvv 3444   class class class wbr 5102   E cep 5530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-eprel 5531  df-xp 5637  df-rel 5638

This theorem is referenced by: (None)