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Mirrors > Home > MPE Home > Th. List > eqeuel | Structured version Visualization version GIF version |
Description: A condition which implies the existence of a unique element of a class. (Contributed by AV, 4-Jan-2022.) |
Ref | Expression |
---|---|
eqeuel | ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)) → ∃!𝑥 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4307 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | 1 | biimpi 215 | . . 3 ⊢ (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴) |
3 | 2 | anim1i 616 | . 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)) → (∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))) |
4 | eleq1w 2817 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
5 | 4 | eu4 2612 | . 2 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦))) |
6 | 3, 5 | sylibr 233 | 1 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)) → ∃!𝑥 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∀wal 1540 ∃wex 1782 ∈ wcel 2107 ∃!weu 2563 ≠ wne 2940 ∅c0 4283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-dif 3914 df-nul 4284 |
This theorem is referenced by: frgr2wwlk1 29315 |
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