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Theorem eueq 3676
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 3675 and eueq 3676. (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 3675 . . 3 ∃*𝑥 𝑥 = 𝐴
21biantru 533 . 2 (∃𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃*𝑥 𝑥 = 𝐴))
3 isset 3483 . 2 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4 df-eu 2654 . 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 1538  wex 1781  wcel 2115  ∃*wmo 2621  ∃!weu 2653  Vcvv 3471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-v 3473
This theorem is referenced by:  eueqi  3677  reuhypd  5293  mptfnf  6456  mptfng  6460  upxp  22207  iotasbc  40938  sprval  43819
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