Theorem epelc 4336

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 4335	. 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 2175  Vcvv 2771   class class class wbr 4043   E cep 4332

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-eprel 4334

This theorem is referenced by:  epel  4337  epini  5050  ecid  6675  ordiso2  7119