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Theorem eqeuel 4276
 Description: A condition which implies the existence of a unique element of a class. (Contributed by AV, 4-Jan-2022.)
Assertion
Ref Expression
eqeuel ((𝐴 ≠ ∅ ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)) → ∃!𝑥 𝑥𝐴)
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem eqeuel
StepHypRef Expression
1 n0 4260 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
21biimpi 219 . . 3 (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴)
32anim1i 617 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)) → (∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)))
4 eleq1w 2872 . . 3 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
54eu4 2676 . 2 (∃!𝑥 𝑥𝐴 ↔ (∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)))
63, 5sylibr 237 1 ((𝐴 ≠ ∅ ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)) → ∃!𝑥 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536  ∃wex 1781   ∈ wcel 2111  ∃!weu 2628   ≠ wne 2987  ∅c0 4243 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-dif 3884  df-nul 4244 This theorem is referenced by:  frgr2wwlk1  28121
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