Theorem bj-epelb 37392

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 5681 available. Check if it is shorter to prove bj-epelg 37391 first or bj-epelb 37392 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 5776	. . . 4 ⊢ Rel E
2	1	brrelex2i 5681	. . 3 ⊢ (𝐴 E 𝐵 → 𝐵 ∈ V)
3	2	pm4.71i 559	. 2 ⊢ (𝐴 E 𝐵 ↔ (𝐴 E 𝐵 ∧ 𝐵 ∈ V))
4		epelg 5525	. . 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 2114  Vcvv 3430   class class class wbr 5086   E cep 5523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-eprel 5524  df-xp 5630  df-rel 5631

This theorem is referenced by: (None)