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Theorem eusn 4450
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 4446 . 2 (∃!𝑥 𝑥𝐴 ↔ ∃𝑥{𝑥𝑥𝐴} = {𝑥})
2 abid2 2925 . . . 4 {𝑥𝑥𝐴} = 𝐴
32eqeq1i 2807 . . 3 ({𝑥𝑥𝐴} = {𝑥} ↔ 𝐴 = {𝑥})
43exbii 1933 . 2 (∃𝑥{𝑥𝑥𝐴} = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥})
51, 4bitri 266 1 (∃!𝑥 𝑥𝐴 ↔ ∃𝑥 𝐴 = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wb 197   = wceq 1637  wex 1859  wcel 2155  ∃!weu 2629  {cab 2788  {csn 4364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-v 3389  df-sn 4365
This theorem is referenced by:  initoid  16853  termoid  16854  initoeu2lem1  16862  funpartfv  32367  irinitoringc  42631
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