Theorem epeli 5547

Description: The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5546. (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 5546	. 2 ⊢ (𝐵 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)

Syntax hints:   ↔ wb 208   ∈ wcel 2141  Vcvv 3453   class class class wbr 5099   E cep 5544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-eprel 5545

This theorem is referenced by:  epel  5548  0sn0ep  5549  smoiso  8328  smoiso2  8335  ecid  8757  ordiso2  9460  cantnflt  9624  cantnfp1lem3  9632  oemapso  9634  cantnflem1b  9638  cantnflem1  9641  cantnf  9645  wemapwe  9649  cnfcomlem  9651  cnfcom  9652  cnfcom3lem  9655  leweon  9964  r0weon  9965  alephiso  10051  fin23lem27  10282  fpwwe2lem8  10593  oniso  28341  ex-eprel  30581  cardpred  35352  vonf1osev  35419  satefvfmla0  35732  satefvfmla1  35739  dftr6  36065  coep  36066  coepr  36067  brsset  36201  brtxpsd  36206  brcart  36244  dfrecs2  36264  dfrdg4  36265  cnambfre  38131  wepwsolem  43583  dnwech  43589  rankrelp  45500