Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-epelb Structured version   Visualization version   GIF version

Theorem bj-epelb 37070
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 5742 available. Check if it is shorter to prove bj-epelg 37069 first or bj-epelb 37070 first. (Contributed by BJ, 14-Jul-2023.)
Assertion
Ref Expression
bj-epelb (𝐴 E 𝐵 ↔ (𝐴𝐵𝐵 ∈ V))

Proof of Theorem bj-epelb
StepHypRef Expression
1 rele 5837 . . . 4 Rel E
21brrelex2i 5742 . . 3 (𝐴 E 𝐵𝐵 ∈ V)
32pm4.71i 559 . 2 (𝐴 E 𝐵 ↔ (𝐴 E 𝐵𝐵 ∈ V))
4 epelg 5585 . . 3 (𝐵 ∈ V → (𝐴 E 𝐵𝐴𝐵))
54pm5.32ri 575 . 2 ((𝐴 E 𝐵𝐵 ∈ V) ↔ (𝐴𝐵𝐵 ∈ V))
63, 5bitri 275 1 (𝐴 E 𝐵 ↔ (𝐴𝐵𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2108  Vcvv 3480   class class class wbr 5143   E cep 5583
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-eprel 5584  df-xp 5691  df-rel 5692
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator