Theorem epeli 5462

Description: The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5460. (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 5460	. 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 2110  Vcvv 3494   class class class wbr 5058   E cep 5458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-eprel 5459

This theorem is referenced by:  epel  5463  0sn0ep  5464  epini  5953  smoiso  7993  smoiso2  8000  ecid  8356  ordiso2  8973  cantnflt  9129  cantnfp1lem3  9137  oemapso  9139  cantnflem1b  9143  cantnflem1  9146  cantnf  9150  wemapwe  9154  cnfcomlem  9156  cnfcom  9157  cnfcom3lem  9160  leweon  9431  r0weon  9432  alephiso  9518  fin23lem27  9744  fpwwe2lem9  10054  ex-eprel  28206  satefvfmla0  32660  satefvfmla1  32667  dftr6  32981  coep  32982  coepr  32983  brsset  33345  brtxpsd  33350  brcart  33388  dfrecs2  33406  dfrdg4  33407  cnambfre  34934  wepwsolem  39635  dnwech  39641