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Theorem epeli 5540
Description: The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5539. (Contributed by Scott Fenton, 11-Apr-2012.)
Hypothesis
Ref Expression
epeli.1 𝐵 ∈ V
Assertion
Ref Expression
epeli (𝐴 E 𝐵𝐴𝐵)

Proof of Theorem epeli
StepHypRef Expression
1 epeli.1 . 2 𝐵 ∈ V
2 epelg 5539 . 2 (𝐵 ∈ V → (𝐴 E 𝐵𝐴𝐵))
31, 2ax-mp 5 1 (𝐴 E 𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2107  Vcvv 3446   class class class wbr 5106   E cep 5537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-eprel 5538
This theorem is referenced by:  epel  5541  0sn0ep  5542  smoiso  8309  smoiso2  8316  ecid  8722  ordiso2  9452  cantnflt  9609  cantnfp1lem3  9617  oemapso  9619  cantnflem1b  9623  cantnflem1  9626  cantnf  9630  wemapwe  9634  cnfcomlem  9636  cnfcom  9637  cnfcom3lem  9640  leweon  9948  r0weon  9949  alephiso  10035  fin23lem27  10265  fpwwe2lem8  10575  ex-eprel  29380  cardpred  33697  satefvfmla0  34015  satefvfmla1  34022  dftr6  34327  coep  34328  coepr  34329  brsset  34477  brtxpsd  34482  brcart  34520  dfrecs2  34538  dfrdg4  34539  cnambfre  36129  wepwsolem  41372  dnwech  41378
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