Theorem epeli 5497

Description: The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5496. (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 5496	. 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 2106  Vcvv 3432   class class class wbr 5074   E cep 5494

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-eprel 5495

This theorem is referenced by:  epel  5498  0sn0ep  5499  smoiso  8193  smoiso2  8200  ecid  8571  ordiso2  9274  cantnflt  9430  cantnfp1lem3  9438  oemapso  9440  cantnflem1b  9444  cantnflem1  9447  cantnf  9451  wemapwe  9455  cnfcomlem  9457  cnfcom  9458  cnfcom3lem  9461  leweon  9767  r0weon  9768  alephiso  9854  fin23lem27  10084  fpwwe2lem8  10394  ex-eprel  28797  cardpred  33062  satefvfmla0  33380  satefvfmla1  33387  dftr6  33718  coep  33719  coepr  33720  brsset  34191  brtxpsd  34196  brcart  34234  dfrecs2  34252  dfrdg4  34253  cnambfre  35825  wepwsolem  40867  dnwech  40873