Theorem bj-epelb 37082

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 5671 available. Check if it is shorter to prove bj-epelg 37081 first or bj-epelb 37082 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 5765	. . . 4 ⊢ Rel E
2	1	brrelex2i 5671	. . 3 ⊢ (𝐴 E 𝐵 → 𝐵 ∈ V)
3	2	pm4.71i 559	. 2 ⊢ (𝐴 E 𝐵 ↔ (𝐴 E 𝐵 ∧ 𝐵 ∈ V))
4		epelg 5515	. . 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 2110  Vcvv 3434   class class class wbr 5089   E cep 5513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-br 5090  df-opab 5152  df-eprel 5514  df-xp 5620  df-rel 5621

This theorem is referenced by: (None)