Theorem epel 5314

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 3412	. 2 ⊢ 𝑥 ∈ V
2	1	epeli 5313	1 ⊢ (𝐴 E 𝑥 ↔ 𝐴 ∈ 𝑥)

Syntax hints:   ↔ wb 198   ∈ wcel 2048   class class class wbr 4923   E cep 5309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pr 5180

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-rab 3091  df-v 3411  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4924  df-opab 4986  df-eprel 5310

This theorem is referenced by:  epse  5383  dfepfr  5385  epfrc  5386  wecmpep  5392  wetrep  5393  epweon  7307  smoiso  7796  smoiso2  7803  ordunifi  8555  ordiso2  8766  ordtypelem8  8776  oismo  8791  wofib  8796  dford2  8869  noinfep  8909  oemapso  8931  wemapwe  8946  alephiso  9310  cflim2  9475  fin23lem27  9540  om2uzisoi  13130  bnj219  31612  efrunt  32399  dftr6  32446  dffr5  32449  elpotr  32486  dfon2lem9  32496  dfon2  32497  domep  32498  brsset  32811  dfon3  32814  brbigcup  32820  brapply  32860  brcup  32861  brcap  32862  dfint3  32874  dfssr2  35132