Theorem bj-epelb 38188

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 5713 available. Check if it is shorter to prove bj-epelg 38187 first or bj-epelb 38188 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 5812	. . . 4 ⊢ Rel E
2	1	brrelex2i 5713	. . 3 ⊢ (𝐴 E 𝐵 → 𝐵 ∈ V)
3	2	pm4.71i 569	. 2 ⊢ (𝐴 E 𝐵 ↔ (𝐴 E 𝐵 ∧ 𝐵 ∈ V))
4		epelg 5557	. . 3 ⊢ (𝐵 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
5	4	pm5.32ri 586	. 2 ⊢ ((𝐴 E 𝐵 ∧ 𝐵 ∈ V) ↔ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ V))
6	3, 5	bitri 278	1 ⊢ (𝐴 E 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ V))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2147  Vcvv 3454   class class class wbr 5106   E cep 5555

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-eprel 5556  df-xp 5662  df-rel 5663

This theorem is used by: (None)