Theorem epel 5591

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 3481	. 2 ⊢ 𝑥 ∈ V
2	1	epeli 5590	1 ⊢ (𝐴 E 𝑥 ↔ 𝐴 ∈ 𝑥)

Syntax hints:   ↔ wb 206   ∈ wcel 2105   class class class wbr 5147   E cep 5587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-eprel 5588

This theorem is referenced by:  epse  5670  dfepfr  5672  epfrc  5673  wecmpep  5680  wetrep  5681  dmep  5936  rnep  5939  epweon  7793  epweonALT  7794  smoiso  8400  smoiso2  8407  ordunifi  9323  ordiso2  9552  ordtypelem8  9562  oismo  9577  wofib  9582  dford2  9657  noinfep  9697  oemapso  9719  wemapwe  9734  alephiso  10135  cflim2  10300  fin23lem27  10365  om2uzisoi  13991  om2noseqiso  28322  bnj219  34725  nummin  35083  efrunt  35692  dftr6  35730  dffr5  35733  elpotr  35762  dfon2lem9  35772  dfon2  35773  brsset  35870  dfon3  35873  brbigcup  35879  brapply  35919  brcup  35920  brcap  35921  dfint3  35933  dfssr2  38480  onsupuni  43217  onsupmaxb  43227