Theorem epeli 5544

Description: The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5543. (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 5543	. 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 3446   class class class wbr 5110   E cep 5541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-eprel 5542

This theorem is referenced by:  epel  5545  0sn0ep  5546  smoiso  8313  smoiso2  8320  ecid  8728  ordiso2  9460  cantnflt  9617  cantnfp1lem3  9625  oemapso  9627  cantnflem1b  9631  cantnflem1  9634  cantnf  9638  wemapwe  9642  cnfcomlem  9644  cnfcom  9645  cnfcom3lem  9648  leweon  9956  r0weon  9957  alephiso  10043  fin23lem27  10273  fpwwe2lem8  10583  ex-eprel  29440  cardpred  33783  satefvfmla0  34099  satefvfmla1  34106  dftr6  34410  coep  34411  coepr  34412  brsset  34550  brtxpsd  34555  brcart  34593  dfrecs2  34611  dfrdg4  34612  cnambfre  36199  wepwsolem  41427  dnwech  41433