Theorem epel 4347

Description: The epsilon relation and the membership relation are the same. (Contributed by NM, 13-Aug-1995.)

Assertion

Ref	Expression
epel	⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)

Proof of Theorem epel

Step	Hyp	Ref	Expression
1		vex 2776	. 2 ⊢ 𝑦 ∈ V
2	1	epelc 4346	1 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)

Syntax hints:   ↔ wb 105   class class class wbr 4051   E cep 4342

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-eprel 4344

This theorem is referenced by:  epse  4397  wetrep  4415  ordsoexmid  4618  zfregfr  4630  ordwe  4632  wessep  4634  reg3exmidlemwe  4635  smoiso  6401  nnwetri  7028  ordiso2  7152  frec2uzisod  10574  nnti  16068