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Theorem eueq 3643
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 3642 and eueq 3643. (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 3642 . . 3 ∃*𝑥 𝑥 = 𝐴
21biantru 530 . 2 (∃𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃*𝑥 𝑥 = 𝐴))
3 isset 3445 . 2 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4 df-eu 2569 . 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 396   = wceq 1539  wex 1782  wcel 2106  ∃*wmo 2538  ∃!weu 2568  Vcvv 3432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434
This theorem is referenced by:  eueqi  3644  reuhypd  5342  mptfnf  6568  mptfng  6572  upxp  22774  iotasbc  42037  sprval  44931
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