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Theorem eusn 3796
 Description: Two ways to express "A is a singleton." (Contributed by NM, 30-Oct-2010.)
Assertion
Ref Expression
eusn (∃!x x Ax A = {x})
Distinct variable group:   x,A

Proof of Theorem eusn
StepHypRef Expression
1 euabsn 3792 . 2 (∃!x x Ax{x x A} = {x})
2 abid2 2470 . . . 4 {x x A} = A
32eqeq1i 2360 . . 3 ({x x A} = {x} ↔ A = {x})
43exbii 1582 . 2 (x{x x A} = {x} ↔ x A = {x})
51, 4bitri 240 1 (∃!x x Ax A = {x})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  {cab 2339  {csn 3737 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-sn 3741 This theorem is referenced by: (None)
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