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Theorem eqeuel 4202
 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 4190 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
21biimpi 208 . . 3 (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴)
32anim1i 606 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)) → (∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)))
4 eleq1w 2841 . . 3 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
54eu4 2647 . 2 (∃!𝑥 𝑥𝐴 ↔ (∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)))
63, 5sylibr 226 1 ((𝐴 ≠ ∅ ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)) → ∃!𝑥 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387  ∀wal 1506  ∃wex 1743   ∈ wcel 2051  ∃!weu 2584   ≠ wne 2960  ∅c0 4172 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-dif 3825  df-nul 4173 This theorem is referenced by:  frgr2wwlk1  27878
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