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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
StepHypRef Expression
1 moeq 3606 . . 3 ∃*𝑥 𝑥 = 𝐴
21biantru 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))
52, 3, 43bitr4i 306 1 (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
Colors of variables: wff setvar class
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
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