Theorem epel 5535

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 3446	. 2 ⊢ 𝑥 ∈ V
2	1	epeli 5534	1 ⊢ (𝐴 E 𝑥 ↔ 𝐴 ∈ 𝑥)

Syntax hints:   ↔ wb 206   ∈ wcel 2114   class class class wbr 5100   E cep 5531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-eprel 5532

This theorem is referenced by:  epse  5614  dfepfr  5616  epfrc  5617  wecmpep  5624  wetrep  5625  dmep  5880  rnep  5884  epweon  7730  epweonALT  7731  smoiso  8304  smoiso2  8311  ordunifi  9202  ordiso2  9432  ordtypelem8  9442  oismo  9457  wofib  9462  dford2  9541  noinfep  9581  oemapso  9603  wemapwe  9618  alephiso  10020  cflim2  10185  fin23lem27  10250  om2uzisoi  13889  om2noseqiso  28310  bnj219  34910  nummin  35270  efrunt  35929  dftr6  35967  dffr5  35970  elpotr  35995  dfon2lem9  36005  dfon2  36006  brsset  36103  dfon3  36106  brbigcup  36112  brapply  36152  brcup  36153  brcap  36154  dfint3  36168  dfssr2  38830  onsupuni  43586  onsupmaxb  43596  rankrelp  45316  sswfaxreg  45343  brpermmodel  45359