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Theorem epel 5538
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 𝑥𝐴𝑥)

Proof of Theorem epel
StepHypRef Expression
1 vex 3447 . 2 𝑥 ∈ V
21epeli 5537 1 (𝐴 E 𝑥𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2106   class class class wbr 5103   E cep 5534
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-eprel 5535
This theorem is referenced by:  epse  5614  dfepfr  5616  epfrc  5617  wecmpep  5623  wetrep  5624  dmep  5877  rnep  5880  epweon  7705  epweonALT  7706  smoiso  8304  smoiso2  8311  ordunifi  9233  ordiso2  9447  ordtypelem8  9457  oismo  9472  wofib  9477  dford2  9552  noinfep  9592  oemapso  9614  wemapwe  9629  alephiso  10030  cflim2  10195  fin23lem27  10260  om2uzisoi  13851  bnj219  33214  nummin  33564  efrunt  34153  dftr6  34194  dffr5  34197  elpotr  34226  dfon2lem9  34236  dfon2  34237  brsset  34441  dfon3  34444  brbigcup  34450  brapply  34490  brcup  34491  brcap  34492  dfint3  34504  dfssr2  36928  onsupuni  41501  onsupmaxb  41511
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