Theorem epeli 5583

Description: The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5582. (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 5582	. 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 5149   E cep 5580

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 5300  ax-nul 5307  ax-pr 5428

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 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-eprel 5581

This theorem is referenced by:  epel  5584  0sn0ep  5585  smoiso  8362  smoiso2  8369  ecid  8776  ordiso2  9510  cantnflt  9667  cantnfp1lem3  9675  oemapso  9677  cantnflem1b  9681  cantnflem1  9684  cantnf  9688  wemapwe  9692  cnfcomlem  9694  cnfcom  9695  cnfcom3lem  9698  leweon  10006  r0weon  10007  alephiso  10093  fin23lem27  10323  fpwwe2lem8  10633  ex-eprel  29686  cardpred  34093  satefvfmla0  34409  satefvfmla1  34416  dftr6  34721  coep  34722  coepr  34723  brsset  34861  brtxpsd  34866  brcart  34904  dfrecs2  34922  dfrdg4  34923  cnambfre  36536  wepwsolem  41784  dnwech  41790