Theorem epeli 5601

Description: The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5600. (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 5600	. 2 ⊢ (𝐵 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)

Syntax hints:   ↔ wb 206   ∈ wcel 2108  Vcvv 3488   class class class wbr 5166   E cep 5598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-eprel 5599

This theorem is referenced by:  epel  5602  0sn0ep  5603  smoiso  8418  smoiso2  8425  ecid  8840  ordiso2  9584  cantnflt  9741  cantnfp1lem3  9749  oemapso  9751  cantnflem1b  9755  cantnflem1  9758  cantnf  9762  wemapwe  9766  cnfcomlem  9768  cnfcom  9769  cnfcom3lem  9772  leweon  10080  r0weon  10081  alephiso  10167  fin23lem27  10397  fpwwe2lem8  10707  ex-eprel  30465  cardpred  35066  satefvfmla0  35386  satefvfmla1  35393  dftr6  35713  coep  35714  coepr  35715  brsset  35853  brtxpsd  35858  brcart  35896  dfrecs2  35914  dfrdg4  35915  cnambfre  37628  wepwsolem  42999  dnwech  43005