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Mirrors > Home > MPE Home > Th. List > snelpwiOLD | Structured version Visualization version GIF version |
Description: Obsolete version of snelpwi 5439 as of 17-Jan-2025. (Contributed by NM, 28-May-1995.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
snelpwiOLD | ⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssi 4807 | . 2 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) | |
2 | snex 5427 | . . 3 ⊢ {𝐴} ∈ V | |
3 | 2 | elpw 4602 | . 2 ⊢ ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵) |
4 | 1, 3 | sylibr 233 | 1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ⊆ wss 3939 𝒫 cpw 4598 {csn 4624 |
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 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3465 df-dif 3942 df-un 3944 df-ss 3956 df-nul 4319 df-pw 4600 df-sn 4625 df-pr 4627 |
This theorem is referenced by: (None) |
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