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Theorem eqeuel 4072
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 4062 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
21biimpi 206 . . 3 (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴)
32anim1i 593 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)) → (∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)))
4 eleq1w 2810 . . 3 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
54eu4 2644 . 2 (∃!𝑥 𝑥𝐴 ↔ (∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)))
63, 5sylibr 224 1 ((𝐴 ≠ ∅ ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)) → ∃!𝑥 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1618  wex 1841  wcel 2127  ∃!weu 2595  wne 2920  c0 4046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-v 3330  df-dif 3706  df-nul 4047
This theorem is referenced by:  frgr2wwlk1  27454
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