Theorem clel4 1894

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

Hypothesis

Ref	Expression
clel4.1	|- B e. V

Assertion

Ref	Expression
clel4	|- (A e. B <-> A.x(x = B -> A e. x))

Distinct variable groups:   x,A   x,B

Proof of Theorem clel4

Step	Hyp	Ref	Expression
1		clel4.1	. . 3 |- B e. V
2		eleq2 1535	. . 3 |- (x = B -> (A e. x <-> A e. B))
3	1, 2	ceqsalv 1827	. 2 |- (A.x(x = B -> A e. x) <-> A e. B)
4	3	bicomi 172	1 |- (A e. B <-> A.x(x = B -> A e. x))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   = wceq 956   e. wcel 958  Vcvv 1811

This theorem is referenced by:  intpr 2563  dfiin2 2588

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  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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