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Mirrors > Home > MPE Home > Th. List > snsspw | Structured version Visualization version GIF version |
Description: The singleton of a class is a subset of its power class. (Contributed by NM, 21-Jun-1993.) |
Ref | Expression |
---|---|
snsspw | ⊢ {𝐴} ⊆ 𝒫 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss 4035 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 ⊆ 𝐴) | |
2 | velsn 4646 | . . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
3 | velpw 4609 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
4 | 1, 2, 3 | 3imtr4i 291 | . 2 ⊢ (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝒫 𝐴) |
5 | 4 | ssriv 3980 | 1 ⊢ {𝐴} ⊆ 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 𝒫 cpw 4604 {csn 4630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3463 df-ss 3961 df-pw 4606 df-sn 4631 |
This theorem is referenced by: snexALT 5383 snwf 9834 tsksn 10785 mnusnd 43847 |
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