Theorem epeli 5581

Description: The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5580. (Contributed by Scott Fenton, 11-Apr-2012.)

Hypothesis

Ref	Expression
epeli.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
epeli	⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)

Proof of Theorem epeli

Step	Hyp	Ref	Expression
1		epeli.1	. 2 ⊢ 𝐵 ∈ V
2		epelg 5580	. 2 ⊢ (𝐵 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)

Syntax hints:   ↔ wb 205   ∈ wcel 2107  Vcvv 3475   class class class wbr 5147   E cep 5578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-eprel 5579

This theorem is referenced by:  epel  5582  0sn0ep  5583  smoiso  8357  smoiso2  8364  ecid  8772  ordiso2  9506  cantnflt  9663  cantnfp1lem3  9671  oemapso  9673  cantnflem1b  9677  cantnflem1  9680  cantnf  9684  wemapwe  9688  cnfcomlem  9690  cnfcom  9691  cnfcom3lem  9694  leweon  10002  r0weon  10003  alephiso  10089  fin23lem27  10319  fpwwe2lem8  10629  ex-eprel  29666  cardpred  34031  satefvfmla0  34347  satefvfmla1  34354  dftr6  34659  coep  34660  coepr  34661  brsset  34799  brtxpsd  34804  brcart  34842  dfrecs2  34860  dfrdg4  34861  cnambfre  36474  wepwsolem  41717  dnwech  41723