Theorem epelc 4322

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 4321	. 2 ⊢ (𝐵 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)

Syntax hints:   ↔ wb 105   ∈ wcel 2164  Vcvv 2760   class class class wbr 4029   E cep 4318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-eprel 4320

This theorem is referenced by:  epel  4323  epini  5036  ecid  6652  ordiso2  7094