Theorem eusn 3592

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 3588	. 2 |- ( E! x x e. A <-> E. x { x | x e. A } = { x } )
2		abid2 2258	. . . 4 |- { x | x e. A } = A
3	2	eqeq1i 2145	. . 3 |- ( { x | x e. A } = { x } <-> A = { x } )
4	3	exbii 1584	. 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 1331   E.wex 1468   e. wcel 1480   E!weu 1997   {cab 2123   {csn 3522

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sn 3528

This theorem is referenced by: (None)