Theorem bj-epelb 34385

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 5602 available. Check if it is shorter to prove bj-epelg 34384 first or bj-epelb 34385 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 5692	. . . 4 ⊢ Rel E
2	1	brrelex2i 5602	. . 3 ⊢ (𝐴 E 𝐵 → 𝐵 ∈ V)
3	2	pm4.71i 562	. 2 ⊢ (𝐴 E 𝐵 ↔ (𝐴 E 𝐵 ∧ 𝐵 ∈ V))
4		epelg 5459	. . 3 ⊢ (𝐵 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
5	4	pm5.32ri 578	. 2 ⊢ ((𝐴 E 𝐵 ∧ 𝐵 ∈ V) ↔ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ V))
6	3, 5	bitri 277	1 ⊢ (𝐴 E 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ V))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∈ wcel 2113  Vcvv 3491   class class class wbr 5059   E cep 5457

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pr 5323

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3493  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-br 5060  df-opab 5122  df-eprel 5458  df-xp 5554  df-rel 5555

This theorem is referenced by: (None)