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Theorem eueq 3705
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 3704 and eueq 3705. (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
StepHypRef Expression
1 moeq 3704 . . 3 ∃*𝑥 𝑥 = 𝐴
21biantru 528 . 2 (∃𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃*𝑥 𝑥 = 𝐴))
3 isset 3485 . 2 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4 df-eu 2561 . 2 (∃!𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃*𝑥 𝑥 = 𝐴))
52, 3, 43bitr4i 302 1 (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1539  wex 1779  wcel 2104  ∃*wmo 2530  ∃!weu 2560  Vcvv 3472
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474
This theorem is referenced by:  eueqi  3706  reuhypd  5418  mptfnf  6686  mptfng  6690  upxp  23349  iotasbc  43482  sprval  46447
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