Theorem eqeuel 4300

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 ≠ ∅ ∧ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)) → ∃!𝑥 𝑥 ∈ 𝐴)

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem eqeuel

Step	Hyp	Ref	Expression
1		n0 4288	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2	1	biimpi 217	. . 3 ⊢ (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴)
3	2	anim1i 621	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)) → (∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)))
4		eleq1w 2823	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
5	4	eu4 2619	. 2 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)))
6	3, 5	sylibr 235	1 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)) → ∃!𝑥 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1545  ∃wex 1786   ∈ wcel 2119  ∃!weu 2572   ≠ wne 2935  ∅c0 4268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-dif 3893  df-nul 4269

This theorem is referenced by:  frgr2wwlk1  30424