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Mirrors > Home > ILE Home > Th. List > eueq | GIF version |
Description: Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.) |
Ref | Expression |
---|---|
eueq | ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqtr3 2059 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑥 = 𝑦) | |
2 | 1 | gen2 1339 | . . 3 ⊢ ∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑥 = 𝑦) |
3 | 2 | biantru 286 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑥 = 𝑦))) |
4 | isset 2561 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
5 | eqeq1 2046 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴)) | |
6 | 5 | eu4 1962 | . 2 ⊢ (∃!𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑥 = 𝑦))) |
7 | 3, 4, 6 | 3bitr4i 201 | 1 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1241 = wceq 1243 ∃wex 1381 ∈ wcel 1393 ∃!weu 1900 Vcvv 2557 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-v 2559 |
This theorem is referenced by: eueq1 2713 moeq 2716 mosubt 2718 reuhypd 4203 mptfng 5024 |
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