Theorem eusn 3444

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 3440	. 2 |- ( E! x x e. A <-> E. x { x | x e. A } = { x } )
2		abid2 2158	. . . 4 |- { x | x e. A } = A
3	2	eqeq1i 2047	. . 3 |- ( { x | x e. A } = { x } <-> A = { x } )
4	3	exbii 1496	. 2 |- ( E. x { x | x e. A } = { x } <-> E. x A = { x } )
5	1, 4	bitri 173	1 |- ( E! x x e. A <-> E. x A = { x } )

Syntax hints:   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   E!weu 1900   {cab 2026   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sn 3381

This theorem is referenced by: (None)