Theorem eueq 3607

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 3606 and eueq 3607. (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 3606	. . 3 ⊢ ∃*𝑥 𝑥 = 𝐴
2	1	biantru 533	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃*𝑥 𝑥 = 𝐴))
3		isset 3411	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4		df-eu 2570	. 2 ⊢ (∃!𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃*𝑥 𝑥 = 𝐴))
5	2, 3, 4	3bitr4i 306	1 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1542  ∃wex 1786   ∈ wcel 2114  ∃*wmo 2538  ∃!weu 2569  Vcvv 3398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3400

This theorem is referenced by:  eueqi  3608  reuhypd  5286  mptfnf  6472  mptfng  6476  upxp  22376  iotasbc  41597  sprval  44494