Theorem risset 1692

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 1058	. 2 |- (E.x(x e. B /\ x = A) <-> E.x(x = A /\ x e. B))
2		df-rex 1657	. 2 |- (E.x e. B x = A <-> E.x(x e. B /\ x = A))
3		df-clel 1478	. 2 |- (A e. B <-> E.x(x = A /\ x e. B))
4	1, 2, 3	3bitr4ri 184	1 |- (A e. B <-> E.x e. B x = A)

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 960   e. wcel 962  E.wex 984  E.wrex 1653

This theorem is referenced by:  0el 2308  sucel 3058  qsid 4319  zorn 4814  negeui 5375  receui 5721  zq 6262  cnsscnp 7798

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 967  ax-4 977  ax-5o 979

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 985  df-clel 1478  df-rex 1657

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