Theorem eusn 3663

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 3659	. 2 |- ( E! x x e. A <-> E. x { x | x e. A } = { x } )
2		abid2 2296	. . . 4 |- { x | x e. A } = A
3	2	eqeq1i 2183	. . 3 |- ( { x | x e. A } = { x } <-> A = { x } )
4	3	exbii 1603	. 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 1353   E.wex 1490   E!weu 2024   e. wcel 2146   {cab 2161   {csn 3589

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-sn 3595

This theorem is referenced by: (None)