Theorem eusn 3668

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 3664	. 2 |- ( E! x x e. A <-> E. x { x | x e. A } = { x } )
2		abid2 2298	. . . 4 |- { x | x e. A } = A
3	2	eqeq1i 2185	. . 3 |- ( { x | x e. A } = { x } <-> A = { x } )
4	3	exbii 1605	. 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 1492   E!weu 2026   e. wcel 2148   {cab 2163   {csn 3594

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-sn 3600

This theorem is referenced by: (None)