Theorem epeli 5432

Description: The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5431. (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 5431	. 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 2111  Vcvv 3441   class class class wbr 5030   E cep 5429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-eprel 5430

This theorem is referenced by:  epel  5433  0sn0ep  5434  epini  5926  smoiso  7982  smoiso2  7989  ecid  8345  ordiso2  8963  cantnflt  9119  cantnfp1lem3  9127  oemapso  9129  cantnflem1b  9133  cantnflem1  9136  cantnf  9140  wemapwe  9144  cnfcomlem  9146  cnfcom  9147  cnfcom3lem  9150  leweon  9422  r0weon  9423  alephiso  9509  fin23lem27  9739  fpwwe2lem9  10049  ex-eprel  28218  cardpred  32473  satefvfmla0  32778  satefvfmla1  32785  dftr6  33099  coep  33100  coepr  33101  brsset  33463  brtxpsd  33468  brcart  33506  dfrecs2  33524  dfrdg4  33525  cnambfre  35105  wepwsolem  39986  dnwech  39992