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Theorem snelpwrVD 43592
Description: Virtual deduction proof of snelpwi 5444. (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 5432 . . 3 {𝐴} ∈ V
2 idn1 43335 . . . 4 (   𝐴𝐵   ▶   𝐴𝐵   )
3 snssi 4812 . . . 4 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
42, 3e1a 43388 . . 3 (   𝐴𝐵   ▶   {𝐴} ⊆ 𝐵   )
5 elpwg 4606 . . . 4 ({𝐴} ∈ V → ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵))
65biimprd 247 . . 3 ({𝐴} ∈ V → ({𝐴} ⊆ 𝐵 → {𝐴} ∈ 𝒫 𝐵))
71, 4, 6e01 43452 . 2 (   𝐴𝐵   ▶   {𝐴} ∈ 𝒫 𝐵   )
87in1 43332 1 (𝐴𝐵 → {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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