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Theorem eusn 4735
Description: Two ways to express "𝐴 is a singleton". (Contributed by NM, 30-Oct-2010.)
Assertion
Ref Expression
eusn (∃!𝑥 𝑥𝐴 ↔ ∃𝑥 𝐴 = {𝑥})
Distinct variable group:   𝑥,𝐴

Proof of Theorem eusn
StepHypRef Expression
1 euabsn 4731 . 2 (∃!𝑥 𝑥𝐴 ↔ ∃𝑥{𝑥𝑥𝐴} = {𝑥})
2 abid2 2872 . . . 4 {𝑥𝑥𝐴} = 𝐴
32eqeq1i 2738 . . 3 ({𝑥𝑥𝐴} = {𝑥} ↔ 𝐴 = {𝑥})
43exbii 1851 . 2 (∃𝑥{𝑥𝑥𝐴} = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥})
51, 4bitri 275 1 (∃!𝑥 𝑥𝐴 ↔ ∃𝑥 𝐴 = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wex 1782  wcel 2107  ∃!weu 2563  {cab 2710  {csn 4629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-sn 4630
This theorem is referenced by:  initoid  17951  termoid  17952  initoeu2lem1  17964  funpartfv  34917  irinitoringc  46967  fullthinc  47666  mndtcbas  47707
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