|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > snelpwg | 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.) Put in closed form and avoid ax-nul 5305. (Revised by BJ, 17-Jan-2025.) | 
| Ref | Expression | 
|---|---|
| snelpwg | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | snssg 4782 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)) | |
| 2 | snexg 5434 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) | |
| 3 | elpwg 4602 | . . 3 ⊢ ({𝐴} ∈ V → ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵)) | |
| 4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵)) | 
| 5 | 1, 4 | bitr4d 282 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2107 Vcvv 3479 ⊆ wss 3950 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: snelpwi 5447 snelpw 5449 | 
| Copyright terms: Public domain | W3C validator |