Theorem epel 5565

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 5564	1 ⊢ (𝐴 E 𝑥 ↔ 𝐴 ∈ 𝑥)

Syntax hints:   ↔ wb 209   ∈ wcel 2149   class class class wbr 5113   E cep 5561

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 5261  ax-pr 5405

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 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-eprel 5562

This theorem is referenced by:  epse  5644  dfepfr  5646  epfrc  5647  wecmpep  5654  wetrep  5655  dmep  5914  rnep  5918  xpdifcnvepel  6167  epweon  7773  epweonALT  7774  smoiso  8348  smoiso2  8355  ordunifi  9249  ordiso2  9476  ordtypelem8  9486  oismo  9501  wofib  9506  dford2  9588  noinfep  9628  oemapso  9650  wemapwe  9665  alephiso  10081  cflim2  10246  fin23lem27  10311  om2uzisoi  13989  om2noseqiso  28460  bnj219  35066  nummin  35426  efrunt  36103  dftr6  36141  dffr5  36144  elpotr  36169  dfon2lem9  36179  dfon2  36180  brsset  36277  dfon3  36280  brbigcup  36286  brapply  36326  brcup  36327  brcap  36328  dfint3  36342  dfssr2  39117  onsupuni  43847  onsupmaxb  43857  rankrelp  45560  sswfaxreg  45587  brpermmodel  45603  hashomiso  45625