Theorem snelpwrVD 45108

Description: Virtual deduction proof of snelpwi 5391. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
snelpwrVD	⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵)

Proof of Theorem snelpwrVD

Step	Hyp	Ref	Expression
1		snex 5380	. . 3 ⊢ {𝐴} ∈ V
2		idn1 44852	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
3		snssi 4763	. . . 4 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)
4	2, 3	e1a 44905	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   {𝐴} ⊆ 𝐵   )
5		elpwg 4556	. . . 4 ⊢ ({𝐴} ∈ V → ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵))
6	5	biimprd 248	. . 3 ⊢ ({𝐴} ∈ V → ({𝐴} ⊆ 𝐵 → {𝐴} ∈ 𝒫 𝐵))
7	1, 4, 6	e01 44969	. 2 ⊢ (   𝐴 ∈ 𝐵   ▶   {𝐴} ∈ 𝒫 𝐵   )
8	7	in1 44849	1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3439   ⊆ wss 3900  𝒫 cpw 4553  {csn 4579

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-v 3441  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-pw 4555  df-sn 4580  df-pr 4582  df-vd1 44848

This theorem is referenced by: (None)