Theorem eueq 3681

Description: A class is a set if and only if there exists a unique set equal to it. (Contributed by NM, 25-Nov-1994.) Shorten combined proofs of moeq 3680 and eueq 3681. (Proof shortened by BJ, 24-Sep-2022.)

Assertion

Ref	Expression
eueq	⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem eueq

Step	Hyp	Ref	Expression
1		moeq 3680	. . 3 ⊢ ∃*𝑥 𝑥 = 𝐴
2	1	biantru 529	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃*𝑥 𝑥 = 𝐴))
3		isset 3464	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4		df-eu 2563	. 2 ⊢ (∃!𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃*𝑥 𝑥 = 𝐴))
5	2, 3, 4	3bitr4i 303	1 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃*wmo 2532  ∃!weu 2562  Vcvv 3450

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452

This theorem is referenced by:  eueqi  3682  reuhypd  5376  mptfnf  6655  mptfng  6659  upxp  23516  iotasbc  44401  sprval  47470