Theorem epeli 5591

Description: The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5590. (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 5590	. 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 2106  Vcvv 3478   class class class wbr 5148   E cep 5588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-eprel 5589

This theorem is referenced by:  epel  5592  0sn0ep  5593  smoiso  8401  smoiso2  8408  ecid  8821  ordiso2  9553  cantnflt  9710  cantnfp1lem3  9718  oemapso  9720  cantnflem1b  9724  cantnflem1  9727  cantnf  9731  wemapwe  9735  cnfcomlem  9737  cnfcom  9738  cnfcom3lem  9741  leweon  10049  r0weon  10050  alephiso  10136  fin23lem27  10366  fpwwe2lem8  10676  ex-eprel  30462  cardpred  35083  satefvfmla0  35403  satefvfmla1  35410  dftr6  35731  coep  35732  coepr  35733  brsset  35871  brtxpsd  35876  brcart  35914  dfrecs2  35932  dfrdg4  35933  cnambfre  37655  wepwsolem  43031  dnwech  43037