Theorem clel3 1893

Description: An alternate definition of class membership when the class is a set.

Hypothesis

Ref	Expression
clel3.1	|- B e. V

Assertion

Ref	Expression
clel3	|- (A e. B <-> E.x(x = B /\ A e. x))

Distinct variable groups:   x,A   x,B

Proof of Theorem clel3

Step	Hyp	Ref	Expression
1		clel3.1	. 2 |- B e. V
2		clel3g 1892	. 2 |- (B e. V -> (A e. B <-> E.x(x = B /\ A e. x)))
3	1, 2	ax-mp 7	1 |- (A e. B <-> E.x(x = B /\ A e. x))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  Vcvv 1811

This theorem is referenced by:  unipr 2515

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-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

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