Theorem eusn 3657

Description: Two ways to express " A is a singleton". (Contributed by NM, 30-Oct-2010.)

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 3653	. 2 |- ( E! x x e. A <-> E. x { x | x e. A } = { x } )
2		abid2 2291	. . . 4 |- { x | x e. A } = A
3	2	eqeq1i 2178	. . 3 |- ( { x | x e. A } = { x } <-> A = { x } )
4	3	exbii 1598	. 2 |- ( E. x { x | x e. A } = { x } <-> E. x A = { x } )
5	1, 4	bitri 183	1 |- ( E! x x e. A <-> E. x A = { x } )

Syntax hints:   <-> wb 104   = wceq 1348   E.wex 1485   E!weu 2019   e. wcel 2141   {cab 2156   {csn 3583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sn 3589

This theorem is referenced by: (None)