Theorem eusn 4706

Description: Two ways to express "𝐴 is a singleton". (Contributed by NM, 30-Oct-2010.)

Assertion

Ref	Expression
eusn	⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 𝐴 = {𝑥})

Distinct variable group:   𝑥,𝐴

Proof of Theorem eusn

Step	Hyp	Ref	Expression
1		euabsn 4702	. 2 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥{𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥})
2		abid2 2872	. . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
3	2	eqeq1i 2740	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥} ↔ 𝐴 = {𝑥})
4	3	exbii 1848	. 2 ⊢ (∃𝑥{𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥})
5	1, 4	bitri 275	1 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 𝐴 = {𝑥})

Syntax hints:   ↔ wb 206   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃!weu 2567  {cab 2713  {csn 4601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-sn 4602

This theorem is referenced by:  initoid  18012  termoid  18013  initoeu2lem1  18025  irinitoringc  21438  funpartfv  35909  fullthinc  49284  istermc2  49309  functermceu  49343  mndtcbas  49406