Theorem eusn 3692

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 3688	. 2 |- ( E! x x e. A <-> E. x { x | x e. A } = { x } )
2		abid2 2314	. . . 4 |- { x | x e. A } = A
3	2	eqeq1i 2201	. . 3 |- ( { x | x e. A } = { x } <-> A = { x } )
4	3	exbii 1616	. 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 1364   E.wex 1503   E!weu 2042   e. wcel 2164   {cab 2179   {csn 3618

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-sn 3624

This theorem is referenced by: (None)