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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 2190 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑥 = 𝑦) | |
2 | 1 | gen2 1443 | . . 3 ⊢ ∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑥 = 𝑦) |
3 | 2 | biantru 300 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑥 = 𝑦))) |
4 | isset 2736 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
5 | eqeq1 2177 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴)) | |
6 | 5 | eu4 2081 | . 2 ⊢ (∃!𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑥 = 𝑦))) |
7 | 3, 4, 6 | 3bitr4i 211 | 1 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1346 = wceq 1348 ∃wex 1485 ∃!weu 2019 ∈ wcel 2141 Vcvv 2730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-v 2732 |
This theorem is referenced by: eueq1 2902 moeq 2905 mosubt 2907 reuhypd 4456 mptfng 5323 upxp 13066 |
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