Theorem epel 5562

Description: The membership relation and the membership predicate agree when the "containing" class is a setvar. Definition 1.6 of [Schloeder] p. 1. (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 3467	. 2 ⊢ 𝑥 ∈ V
2	1	epeli 5561	1 ⊢ (𝐴 E 𝑥 ↔ 𝐴 ∈ 𝑥)

Syntax hints:   ↔ wb 209   ∈ wcel 2149   class class class wbr 5110   E cep 5558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-eprel 5559

This theorem is referenced by:  epse  5641  dfepfr  5643  epfrc  5644  wecmpep  5651  wetrep  5652  dmep  5911  rnep  5915  xpdifcnvepel  6164  epweon  7770  epweonALT  7771  smoiso  8345  smoiso2  8352  ordunifi  9246  ordiso2  9473  ordtypelem8  9483  oismo  9498  wofib  9503  dford2  9585  noinfep  9625  oemapso  9647  wemapwe  9662  alephiso  10078  cflim2  10243  fin23lem27  10308  om2uzisoi  13986  om2noseqiso  28457  bnj219  35063  nummin  35423  efrunt  36100  dftr6  36138  dffr5  36141  elpotr  36166  dfon2lem9  36176  dfon2  36177  brsset  36274  dfon3  36277  brbigcup  36283  brapply  36323  brcup  36324  brcap  36325  dfint3  36339  dfssr2  39113  onsupuni  43843  onsupmaxb  43853  rankrelp  45556  sswfaxreg  45583  brpermmodel  45599  hashomiso  45621