Theorem epeli 5555

Description: The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5554. (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 5554	. 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 3459   class class class wbr 5119   E cep 5552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-eprel 5553

This theorem is referenced by:  epel  5556  0sn0ep  5557  smoiso  8376  smoiso2  8383  ecid  8796  ordiso2  9529  cantnflt  9686  cantnfp1lem3  9694  oemapso  9696  cantnflem1b  9700  cantnflem1  9703  cantnf  9707  wemapwe  9711  cnfcomlem  9713  cnfcom  9714  cnfcom3lem  9717  leweon  10025  r0weon  10026  alephiso  10112  fin23lem27  10342  fpwwe2lem8  10652  onsiso  28221  ex-eprel  30414  cardpred  35121  satefvfmla0  35440  satefvfmla1  35447  dftr6  35768  coep  35769  coepr  35770  brsset  35907  brtxpsd  35912  brcart  35950  dfrecs2  35968  dfrdg4  35969  cnambfre  37692  wepwsolem  43066  dnwech  43072  rankrelp  44985