Theorem epeli 5561

Description: The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5560. (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 5560	. 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 3453   class class class wbr 5107   E cep 5558

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 2734  ax-sep 5255  ax-pr 5402

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 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-eprel 5559

This theorem is used by:  epel  5562  0sn0ep  5563  smoiso  8355  smoiso2  8362  ecid  8784  ordiso2  9491  cantnflt  9655  cantnfp1lem3  9663  oemapso  9665  cantnflem1b  9669  cantnflem1  9672  cantnf  9676  wemapwe  9680  cnfcomlem  9682  cnfcom  9683  cnfcom3lem  9686  leweon  10018  r0weon  10019  alephiso  10105  fin23lem27  10334  fpwwe2lem8  10651  oniso  28544  ex-eprel  30921  cardpred  35605  vonf1osev  35717  satefvfmla0  36005  satefvfmla1  36012  dftr6  36338  coep  36339  coepr  36340  brsset  36474  brtxpsd  36479  brcart  36517  dfrecs2  36537  dfrdg4  36538  cnambfre  38425  wepwsolem  43891  dnwech  43897  rankrelp  45791