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Theorem eusn 4409
 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 4405 . 2 (∃!𝑥 𝑥𝐴 ↔ ∃𝑥{𝑥𝑥𝐴} = {𝑥})
2 abid2 2883 . . . 4 {𝑥𝑥𝐴} = 𝐴
32eqeq1i 2765 . . 3 ({𝑥𝑥𝐴} = {𝑥} ↔ 𝐴 = {𝑥})
43exbii 1923 . 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 1632  ∃wex 1853   ∈ wcel 2139  ∃!weu 2607  {cab 2746  {csn 4321 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-sn 4322 This theorem is referenced by:  initoid  16876  termoid  16877  initoeu2lem1  16885  funpartfv  32379  irinitoringc  42597
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