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Mirrors > Home > ILE Home > Th. List > eueq1 | GIF version |
Description: Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.) |
Ref | Expression |
---|---|
eueq1.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
eueq1 | ⊢ ∃!𝑥 𝑥 = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eueq1.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | eueq 2859 | . 2 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴) | |
3 | 1, 2 | mpbi 144 | 1 ⊢ ∃!𝑥 𝑥 = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 ∃!weu 2000 Vcvv 2689 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-v 2691 |
This theorem is referenced by: eueq2dc 2861 eueq3dc 2862 fsn 5600 |
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