Theorem epeli 5557

Description: The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5556. (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 5556	. 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 2145  Vcvv 3450   class class class wbr 5103   E cep 5554

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 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

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 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-eprel 5555

This theorem is used by:  epel  5558  0sn0ep  5559  smoiso  8351  smoiso2  8358  ecid  8780  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  10647  oniso  28536  ex-eprel  30913  cardpred  35597  vonf1osev  35709  satefvfmla0  35997  satefvfmla1  36004  dftr6  36330  coep  36331  coepr  36332  brsset  36466  brtxpsd  36471  brcart  36509  dfrecs2  36529  dfrdg4  36530  cnambfre  38417  wepwsolem  43883  dnwech  43889  rankrelp  45783