Theorem epelc 4951

Description: The epsilon relationship and the membership relation are the same. (Contributed by Scott Fenton, 11-Apr-2012.)

Hypothesis

Ref	Expression
epelc.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
epelc	⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)

Proof of Theorem epelc

Step	Hyp	Ref	Expression
1		epelc.1	. 2 ⊢ 𝐵 ∈ V
2		epelg 4950	. 2 ⊢ (𝐵 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)

Syntax hints:   ↔ wb 195   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583   E cep 4947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-eprel 4949

This theorem is referenced by:  epel  4952  epini  5414  smoiso  7346  smoiso2  7353  ecid  7699  ordiso2  8303  oismo  8328  cantnflt  8452  cantnfp1lem3  8460  oemapso  8462  cantnflem1b  8466  cantnflem1  8469  cantnf  8473  wemapwe  8477  cnfcomlem  8479  cnfcom  8480  cnfcom3lem  8483  leweon  8717  r0weon  8718  alephiso  8804  fin23lem27  9033  fpwwe2lem9  9339  ex-eprel  26682  dftr6  30893  coep  30894  coepr  30895  brsset  31166  brtxpsd  31171  brcart  31209  dfrecs2  31227  dfrdg4  31228  cnambfre  32628  wepwsolem  36630  dnwech  36636