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Mirrors > Home > MPE Home > Th. List > Mathboxes > snelpwrVD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of snelpwi 5329. (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 5324 | . . 3 ⊢ {𝐴} ∈ V | |
2 | idn1 40901 | . . . 4 ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 ) | |
3 | snssi 4735 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) | |
4 | 2, 3 | e1a 40954 | . . 3 ⊢ ( 𝐴 ∈ 𝐵 ▶ {𝐴} ⊆ 𝐵 ) |
5 | elpwg 4545 | . . . 4 ⊢ ({𝐴} ∈ V → ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵)) | |
6 | 5 | biimprd 250 | . . 3 ⊢ ({𝐴} ∈ V → ({𝐴} ⊆ 𝐵 → {𝐴} ∈ 𝒫 𝐵)) |
7 | 1, 4, 6 | e01 41018 | . 2 ⊢ ( 𝐴 ∈ 𝐵 ▶ {𝐴} ∈ 𝒫 𝐵 ) |
8 | 7 | in1 40898 | 1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 Vcvv 3495 ⊆ wss 3936 𝒫 cpw 4539 {csn 4561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-pw 4541 df-sn 4562 df-pr 4564 df-vd1 40897 |
This theorem is referenced by: (None) |
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