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Mirrors > Home > MPE Home > Th. List > eueq1 | Structured version Visualization version 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 3345 | . 2 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴) | |
3 | 1, 2 | mpbi 219 | 1 ⊢ ∃!𝑥 𝑥 = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ∃!weu 2458 Vcvv 3173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-v 3175 |
This theorem is referenced by: eueq2 3347 eueq3 3348 fsn 6308 bj-nuliota 32210 |
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