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Theorem snelpwrVD 43674
Description: Virtual deduction proof of snelpwi 5443. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snelpwrVD (𝐴𝐵 → {𝐴} ∈ 𝒫 𝐵)

Proof of Theorem snelpwrVD
StepHypRef Expression
1 snex 5431 . . 3 {𝐴} ∈ V
2 idn1 43417 . . . 4 (   𝐴𝐵   ▶   𝐴𝐵   )
3 snssi 4811 . . . 4 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
42, 3e1a 43470 . . 3 (   𝐴𝐵   ▶   {𝐴} ⊆ 𝐵   )
5 elpwg 4605 . . . 4 ({𝐴} ∈ V → ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵))
65biimprd 247 . . 3 ({𝐴} ∈ V → ({𝐴} ⊆ 𝐵 → {𝐴} ∈ 𝒫 𝐵))
71, 4, 6e01 43534 . 2 (   𝐴𝐵   ▶   {𝐴} ∈ 𝒫 𝐵   )
87in1 43414 1 (𝐴𝐵 → {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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