Theorem eusn 3797

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 3793	. 2 ⊢ (∃!x x ∈ A ↔ ∃x{x ∣ x ∈ A} = {x})
2		abid2 2471	. . . 4 ⊢ {x ∣ x ∈ A} = A
3	2	eqeq1i 2360	. . 3 ⊢ ({x ∣ x ∈ A} = {x} ↔ A = {x})
4	3	exbii 1582	. 2 ⊢ (∃x{x ∣ x ∈ A} = {x} ↔ ∃x A = {x})
5	1, 4	bitri 240	1 ⊢ (∃!x x ∈ A ↔ ∃x A = {x})

Syntax hints:   ↔ wb 176  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  {cab 2339  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-sn 3742

This theorem is referenced by: (None)