Theorem eusn 3096

Description: Two ways to express "A is a singleton."

Assertion

Ref	Expression
eusn	|- (E!x x e. A <-> E.x A = {x})

Distinct variable group:   x,A

Proof of Theorem eusn

Step	Hyp	Ref	Expression
1		euabsn 3095	. 2 |- (E!x x e. A <-> E.x{x | x e. A} = {x})
2		abid2 2011	. . . 4 |- {x | x e. A} = A
3	2	eqeq1i 1891	. . 3 |- ({x | x e. A} = {x} <-> A = {x})
4	3	exbii 1398	. 2 |- (E.x{x | x e. A} = {x} <-> E.x A = {x})
5	1, 4	bitri 190	1 |- (E!x x e. A <-> E.x A = {x})

Syntax hints:   <-> wb 163   = wceq 1298   e. wcel 1300  E.wex 1326  E!weu 1771  {cab 1871  {csn 3044

This theorem is referenced by:  eufromeq6 3833

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-clab 1872  df-cleq 1877  df-clel 1880  df-sn 3049

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