Theorem epel 5602

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 3492	. 2 ⊢ 𝑥 ∈ V
2	1	epeli 5601	1 ⊢ (𝐴 E 𝑥 ↔ 𝐴 ∈ 𝑥)

Syntax hints:   ↔ wb 206   ∈ wcel 2108   class class class wbr 5166   E cep 5598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-eprel 5599

This theorem is referenced by:  epse  5682  dfepfr  5684  epfrc  5685  wecmpep  5692  wetrep  5693  dmep  5948  rnep  5951  epweon  7810  epweonALT  7811  smoiso  8418  smoiso2  8425  ordunifi  9354  ordiso2  9584  ordtypelem8  9594  oismo  9609  wofib  9614  dford2  9689  noinfep  9729  oemapso  9751  wemapwe  9766  alephiso  10167  cflim2  10332  fin23lem27  10397  om2uzisoi  14005  om2noseqiso  28326  bnj219  34709  nummin  35067  efrunt  35675  dftr6  35713  dffr5  35716  elpotr  35745  dfon2lem9  35755  dfon2  35756  brsset  35853  dfon3  35856  brbigcup  35862  brapply  35902  brcup  35903  brcap  35904  dfint3  35916  dfssr2  38455  onsupuni  43190  onsupmaxb  43200