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Theorem eusn 4407
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 4403 . 2 (∃!𝑥 𝑥𝐴 ↔ ∃𝑥{𝑥𝑥𝐴} = {𝑥})
2 abid2 2881 . . . 4 {𝑥𝑥𝐴} = 𝐴
32eqeq1i 2763 . . 3 ({𝑥𝑥𝐴} = {𝑥} ↔ 𝐴 = {𝑥})
43exbii 1921 . 2 (∃𝑥{𝑥𝑥𝐴} = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥})
51, 4bitri 264 1 (∃!𝑥 𝑥𝐴 ↔ ∃𝑥 𝐴 = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1630  wex 1851  wcel 2137  ∃!weu 2605  {cab 2744  {csn 4319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-v 3340  df-sn 4320
This theorem is referenced by:  initoid  16854  termoid  16855  initoeu2lem1  16863  funpartfv  32356  irinitoringc  42577
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