![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > eueqi | Structured version Visualization version GIF version |
Description: There exists a unique set equal to a given set. Inference associated with euequ 2586. See euequ 2586 in the case of a setvar. (Contributed by NM, 5-Apr-1995.) |
Ref | Expression |
---|---|
eueqi.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
eueqi | ⊢ ∃!𝑥 𝑥 = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eueqi.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | eueq 3701 | . 2 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴) | |
3 | 1, 2 | mpbi 229 | 1 ⊢ ∃!𝑥 𝑥 = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ∃!weu 2557 Vcvv 3469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-v 3471 |
This theorem is referenced by: eueq2 3703 eueq3 3704 fsn 7138 bj-nuliota 36526 prprval 46826 |
Copyright terms: Public domain | W3C validator |