Theorem eusn 3644

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 3640	. 2 |- ( E! x x e. A <-> E. x { x | x e. A } = { x } )
2		abid2 2285	. . . 4 |- { x | x e. A } = A
3	2	eqeq1i 2172	. . 3 |- ( { x | x e. A } = { x } <-> A = { x } )
4	3	exbii 1592	. 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 1342   E.wex 1479   E!weu 2013   e. wcel 2135   {cab 2150   {csn 3570

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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-sn 3576

This theorem is referenced by: (None)