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Theorem eueq 3696
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 3695 and eueq 3696. (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 3695 . . 3 ∃*𝑥 𝑥 = 𝐴
21biantru 529 . 2 (∃𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃*𝑥 𝑥 = 𝐴))
3 isset 3479 . 2 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4 df-eu 2555 . 2 (∃!𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃*𝑥 𝑥 = 𝐴))
52, 3, 43bitr4i 303 1 (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wex 1773  wcel 2098  ∃*wmo 2524  ∃!weu 2554  Vcvv 3466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468
This theorem is referenced by:  eueqi  3697  reuhypd  5407  mptfnf  6675  mptfng  6679  upxp  23449  iotasbc  43667  sprval  46632
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