Theorem epel 5448

Description: The membership relation and the membership predicate agree when the "containing" class is a setvar. (Contributed by NM, 13-Aug-1995.) Replace the first setvar variable with a class variable. (Revised by BJ, 13-Sep-2022.)

Assertion

Ref	Expression
epel	⊢ (𝐴 E 𝑥 ↔ 𝐴 ∈ 𝑥)

Proof of Theorem epel

Step	Hyp	Ref	Expression
1		vex 3402	. 2 ⊢ 𝑥 ∈ V
2	1	epeli 5447	1 ⊢ (𝐴 E 𝑥 ↔ 𝐴 ∈ 𝑥)

Syntax hints:   ↔ wb 209   ∈ wcel 2112   class class class wbr 5039   E cep 5444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ne 2933  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-br 5040  df-opab 5102  df-eprel 5445

This theorem is referenced by:  epse  5519  dfepfr  5521  epfrc  5522  wecmpep  5528  wetrep  5529  dmep  5777  domepOLD  5778  rnep  5781  epweon  7538  smoiso  8077  smoiso2  8084  ordunifi  8899  ordiso2  9109  ordtypelem8  9119  oismo  9134  wofib  9139  dford2  9213  noinfep  9253  oemapso  9275  wemapwe  9290  alephiso  9677  cflim2  9842  fin23lem27  9907  om2uzisoi  13492  bnj219  32378  nummin  32730  efrunt  33321  dftr6  33387  dffr5  33390  elpotr  33427  dfon2lem9  33437  dfon2  33438  brsset  33877  dfon3  33880  brbigcup  33886  brapply  33926  brcup  33927  brcap  33928  dfint3  33940  dfssr2  36303