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| Mirrors > Home > MPE Home > Th. List > snelpw | Structured version Visualization version GIF version | ||
| Description: A singleton of a set is a member of the powerclass of a class if and only if that set is a member of that class. (Contributed by NM, 1-Apr-1998.) | 
| Ref | Expression | 
|---|---|
| snelpw.ex | ⊢ 𝐴 ∈ V | 
| Ref | Expression | 
|---|---|
| snelpw | ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | snelpw.ex | . 2 ⊢ 𝐴 ∈ V | |
| 2 | snelpwg 5446 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∈ wcel 2107 Vcvv 3479 𝒫 cpw 4599 {csn 4625 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-un 3955 df-ss 3967 df-pw 4601 df-sn 4626 df-pr 4628 | 
| This theorem is referenced by: pwfir 9356 dis2ndc 23469 dislly 23506 n0scut 28339 n0ons 28340 | 
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