Theorem eusn 3749

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 3745	. 2 |- ( E! x x e. A <-> E. x { x | x e. A } = { x } )
2		abid2 2353	. . . 4 |- { x | x e. A } = A
3	2	eqeq1i 2239	. . 3 |- ( { x | x e. A } = { x } <-> A = { x } )
4	3	exbii 1654	. 2 |- ( E. x { x | x e. A } = { x } <-> E. x A = { x } )
5	1, 4	bitri 184	1 |- ( E! x x e. A <-> E. x A = { x } )

Syntax hints:   <-> wb 105   = wceq 1398   E.wex 1541   E!weu 2079   e. wcel 2202   {cab 2217   {csn 3673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-sn 3679

This theorem is referenced by: (None)