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Theorem eusn 4208
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 4204 . 2 (∃!𝑥 𝑥𝐴 ↔ ∃𝑥{𝑥𝑥𝐴} = {𝑥})
2 abid2 2731 . . . 4 {𝑥𝑥𝐴} = 𝐴
32eqeq1i 2614 . . 3 ({𝑥𝑥𝐴} = {𝑥} ↔ 𝐴 = {𝑥})
43exbii 1763 . 2 (∃𝑥{𝑥𝑥𝐴} = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥})
51, 4bitri 262 1 (∃!𝑥 𝑥𝐴 ↔ ∃𝑥 𝐴 = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wb 194   = wceq 1474  wex 1694  wcel 1976  ∃!weu 2457  {cab 2595  {csn 4124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174  df-sn 4125
This theorem is referenced by:  initoid  16426  termoid  16427  initoeu2lem1  16435  funpartfv  31015  irinitoringc  41842
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