Theorem epel 5584

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 3479	. 2 ⊢ 𝑥 ∈ V
2	1	epeli 5583	1 ⊢ (𝐴 E 𝑥 ↔ 𝐴 ∈ 𝑥)

Syntax hints:   ↔ wb 205   ∈ wcel 2107   class class class wbr 5149   E cep 5580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-eprel 5581

This theorem is referenced by:  epse  5660  dfepfr  5662  epfrc  5663  wecmpep  5669  wetrep  5670  dmep  5924  rnep  5927  epweon  7762  epweonALT  7763  smoiso  8362  smoiso2  8369  ordunifi  9293  ordiso2  9510  ordtypelem8  9520  oismo  9535  wofib  9540  dford2  9615  noinfep  9655  oemapso  9677  wemapwe  9692  alephiso  10093  cflim2  10258  fin23lem27  10323  om2uzisoi  13919  bnj219  33744  nummin  34094  efrunt  34682  dftr6  34721  dffr5  34724  elpotr  34753  dfon2lem9  34763  dfon2  34764  brsset  34861  dfon3  34864  brbigcup  34870  brapply  34910  brcup  34911  brcap  34912  dfint3  34924  dfssr2  37369  onsupuni  41978  onsupmaxb  41988