Theorem epelc 2833

Description: The epsilon relation and the membership relation are the same.

Hypotheses

Ref	Expression
epelc.1	|- A e. V
epelc.2	|- B e. V

Assertion

Ref	Expression
epelc	|- (AEB <-> A e. B)

Proof of Theorem epelc

Step	Hyp	Ref	Expression
1		epelc.1	. 2 |- A e. V
2		epelc.2	. 2 |- B e. V
3		eleq1 1534	. 2 |- (x = A -> (x e. y <-> A e. y))
4		eleq2 1535	. 2 |- (y = B -> (A e. y <-> A e. B))
5		df-eprel 2832	. 2 |- E = {<.x, y>. | x e. y}
6	1, 2, 3, 4, 5	brab 2821	1 |- (AEB <-> A e. B)

Syntax hints:   <-> wb 146   e. wcel 958  Vcvv 1811   class class class wbr 2619  Ecep 2830

This theorem is referenced by:  epel 2834  ecid 4300  alephiso 4892

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-eprel 2832

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