Theorem risset 1682

Description: Two ways to say "A belongs to B."

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem risset

Step	Hyp	Ref	Expression
1		exancom 1052	. 2 |- (E.x(x e. B /\ x = A) <-> E.x(x = A /\ x e. B))
2		df-rex 1647	. 2 |- (E.x e. B x = A <-> E.x(x e. B /\ x = A))
3		df-clel 1470	. 2 |- (A e. B <-> E.x(x = A /\ x e. B))
4	1, 2, 3	3bitr4r 184	1 |- (A e. B <-> E.x e. B x = A)

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978  E.wrex 1643

This theorem is referenced by:  0el 2292  sucel 3037  qsid 4291  zorn 4777  negeu 5335  receu 5678  zqt 6206  cnsscnp 7722

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-4 971  ax-5o 973

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-clel 1470  df-rex 1647

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