Theorem epeli 5565

Description: The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5564. (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 5564	. 2 ⊢ (𝐵 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)

Syntax hints:   ↔ wb 209   ∈ wcel 2143  Vcvv 3455   class class class wbr 5110   E cep 5562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-eprel 5563

This theorem is referenced by:  epel  5566  0sn0ep  5567  smoiso  8350  smoiso2  8357  ecid  8779  ordiso2  9478  cantnflt  9642  cantnfp1lem3  9650  oemapso  9652  cantnflem1b  9656  cantnflem1  9659  cantnf  9663  wemapwe  9667  cnfcomlem  9669  cnfcom  9670  cnfcom3lem  9673  leweon  9996  r0weon  9997  alephiso  10083  fin23lem27  10313  fpwwe2lem8  10624  oniso  28442  ex-eprel  30762  cardpred  35461  vonf1osev  35574  satefvfmla0  35888  satefvfmla1  35895  dftr6  36221  coep  36222  coepr  36223  brsset  36357  brtxpsd  36362  brcart  36400  dfrecs2  36420  dfrdg4  36421  cnambfre  38297  wepwsolem  43749  dnwech  43755  rankrelp  45649