Theorem eusn 3696

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 3692	. 2 |- ( E! x x e. A <-> E. x { x | x e. A } = { x } )
2		abid2 2317	. . . 4 |- { x | x e. A } = A
3	2	eqeq1i 2204	. . 3 |- ( { x | x e. A } = { x } <-> A = { x } )
4	3	exbii 1619	. 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 1506   E!weu 2045   e. wcel 2167   {cab 2182   {csn 3622

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sn 3628

This theorem is referenced by: (None)