Theorem eueq 3674

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 3673 and eueq 3674. (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 3673	. . 3 ⊢ ∃*𝑥 𝑥 = 𝐴
2	1	biantru 538	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃*𝑥 𝑥 = 𝐴))
3		isset 3471	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4		df-eu 2599	. 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 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∃*wmo 2567  ∃!weu 2598  Vcvv 3457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459

This theorem is referenced by:  eueqi  3675  reuhypd  5381  mptfnf  6660  mptfng  6664  upxp  23741  iotasbc  44993  sprval  48083