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Theorem eqeuel 4357
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 4341 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
21biimpi 215 . . 3 (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴)
32anim1i 614 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)) → (∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)))
4 eleq1w 2810 . . 3 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
54eu4 2605 . 2 (∃!𝑥 𝑥𝐴 ↔ (∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)))
63, 5sylibr 233 1 ((𝐴 ≠ ∅ ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)) → ∃!𝑥 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1531  wex 1773  wcel 2098  ∃!weu 2556  wne 2934  c0 4317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-dif 3946  df-nul 4318
This theorem is referenced by:  frgr2wwlk1  30087
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