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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 5444. (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 5432 | . . 3 ⊢ {𝐴} ∈ V | |
2 | idn1 43335 | . . . 4 ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 ) | |
3 | snssi 4812 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) | |
4 | 2, 3 | e1a 43388 | . . 3 ⊢ ( 𝐴 ∈ 𝐵 ▶ {𝐴} ⊆ 𝐵 ) |
5 | elpwg 4606 | . . . 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 43452 | . 2 ⊢ ( 𝐴 ∈ 𝐵 ▶ {𝐴} ∈ 𝒫 𝐵 ) |
8 | 7 | in1 43332 | 1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3475 ⊆ wss 3949 𝒫 cpw 4603 {csn 4629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-pw 4605 df-sn 4630 df-pr 4632 df-vd1 43331 |
This theorem is referenced by: (None) |
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