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Mathbox for Alan Sare |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > snelpwrVD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of snelpwi 5443. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
snelpwrVD | ⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5431 | . . 3 ⊢ {𝐴} ∈ V | |
2 | idn1 43417 | . . . 4 ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 ) | |
3 | snssi 4811 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) | |
4 | 2, 3 | e1a 43470 | . . 3 ⊢ ( 𝐴 ∈ 𝐵 ▶ {𝐴} ⊆ 𝐵 ) |
5 | elpwg 4605 | . . . 4 ⊢ ({𝐴} ∈ V → ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵)) | |
6 | 5 | biimprd 247 | . . 3 ⊢ ({𝐴} ∈ V → ({𝐴} ⊆ 𝐵 → {𝐴} ∈ 𝒫 𝐵)) |
7 | 1, 4, 6 | e01 43534 | . 2 ⊢ ( 𝐴 ∈ 𝐵 ▶ {𝐴} ∈ 𝒫 𝐵 ) |
8 | 7 | in1 43414 | 1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3948 𝒫 cpw 4602 {csn 4628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-pw 4604 df-sn 4629 df-pr 4631 df-vd1 43413 |
This theorem is referenced by: (None) |
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