Theorem epel 5471

Description: The membership relation and the membership predicate agree when the "containing" class is a setvar. (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 3499	. 2 ⊢ 𝑥 ∈ V
2	1	epeli 5470	1 ⊢ (𝐴 E 𝑥 ↔ 𝐴 ∈ 𝑥)

Syntax hints:   ↔ wb 208   ∈ wcel 2114   class class class wbr 5068   E cep 5466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-eprel 5467

This theorem is referenced by:  epse  5540  dfepfr  5542  epfrc  5543  wecmpep  5549  wetrep  5550  dmep  5795  domepOLD  5796  rnep  5799  epweon  7499  smoiso  8001  smoiso2  8008  ordunifi  8770  ordiso2  8981  ordtypelem8  8991  oismo  9006  wofib  9011  dford2  9085  noinfep  9125  oemapso  9147  wemapwe  9162  alephiso  9526  cflim2  9687  fin23lem27  9752  om2uzisoi  13325  bnj219  32005  efrunt  32941  dftr6  32988  dffr5  32991  elpotr  33028  dfon2lem9  33038  dfon2  33039  brsset  33352  dfon3  33355  brbigcup  33361  brapply  33401  brcup  33402  brcap  33403  dfint3  33415  dfssr2  35741