Theorem epeli 5521

Description: The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5520. (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 5520	. 2 ⊢ (𝐵 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)

Syntax hints:   ↔ wb 206   ∈ wcel 2113  Vcvv 3437   class class class wbr 5093   E cep 5518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-eprel 5519

This theorem is referenced by:  epel  5522  0sn0ep  5523  smoiso  8288  smoiso2  8295  ecid  8710  ordiso2  9408  cantnflt  9569  cantnfp1lem3  9577  oemapso  9579  cantnflem1b  9583  cantnflem1  9586  cantnf  9590  wemapwe  9594  cnfcomlem  9596  cnfcom  9597  cnfcom3lem  9600  leweon  9909  r0weon  9910  alephiso  9996  fin23lem27  10226  fpwwe2lem8  10536  onsiso  28206  ex-eprel  30415  cardpred  35124  satefvfmla0  35483  satefvfmla1  35490  dftr6  35816  coep  35817  coepr  35818  brsset  35952  brtxpsd  35957  brcart  35995  dfrecs2  36015  dfrdg4  36016  cnambfre  37728  wepwsolem  43159  dnwech  43165  rankrelp  45077