Theorem epel 5517

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 3440	. 2 ⊢ 𝑥 ∈ V
2	1	epeli 5516	1 ⊢ (𝐴 E 𝑥 ↔ 𝐴 ∈ 𝑥)

Syntax hints:   ↔ wb 206   ∈ wcel 2111   class class class wbr 5089   E cep 5513

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-eprel 5514

This theorem is referenced by:  epse  5596  dfepfr  5598  epfrc  5599  wecmpep  5606  wetrep  5607  dmep  5862  rnep  5866  epweon  7708  epweonALT  7709  smoiso  8282  smoiso2  8289  ordunifi  9174  ordiso2  9401  ordtypelem8  9411  oismo  9426  wofib  9431  dford2  9510  noinfep  9550  oemapso  9572  wemapwe  9587  alephiso  9989  cflim2  10154  fin23lem27  10219  om2uzisoi  13861  om2noseqiso  28232  bnj219  34745  nummin  35104  efrunt  35757  dftr6  35795  dffr5  35798  elpotr  35823  dfon2lem9  35833  dfon2  35834  brsset  35931  dfon3  35934  brbigcup  35940  brapply  35980  brcup  35981  brcap  35982  dfint3  35996  dfssr2  38590  onsupuni  43321  onsupmaxb  43331  rankrelp  45052  sswfaxreg  45079  brpermmodel  45095