Theorem epeli 5568

Description: The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5567. (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 5567	. 2 ⊢ (𝐵 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2146  Vcvv 3458   class class class wbr 5114   E cep 5565

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-eprel 5566

This theorem is used by:  epel  5569  0sn0ep  5570  smoiso  8358  smoiso2  8365  ecid  8787  ordiso2  9487  cantnflt  9651  cantnfp1lem3  9659  oemapso  9661  cantnflem1b  9665  cantnflem1  9668  cantnf  9672  wemapwe  9676  cnfcomlem  9678  cnfcom  9679  cnfcom3lem  9682  leweon  10014  r0weon  10015  alephiso  10101  fin23lem27  10330  fpwwe2lem8  10641  oniso  28501  ex-eprel  30821  cardpred  35508  vonf1osev  35620  satefvfmla0  35931  satefvfmla1  35938  dftr6  36264  coep  36265  coepr  36266  brsset  36400  brtxpsd  36405  brcart  36443  dfrecs2  36463  dfrdg4  36464  cnambfre  38360  wepwsolem  43810  dnwech  43816  rankrelp  45710