Theorem snelpwrVD 39380

Description: Virtual deduction proof of snelpwi 4942. (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 4938	. . 3 ⊢ {𝐴} ∈ V
2		idn1 39107	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
3		snssi 4371	. . . 4 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)
4	2, 3	e1a 39169	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   {𝐴} ⊆ 𝐵   )
5		elpwg 4199	. . . 4 ⊢ ({𝐴} ∈ V → ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵))
6	5	biimprd 238	. . 3 ⊢ ({𝐴} ∈ V → ({𝐴} ⊆ 𝐵 → {𝐴} ∈ 𝒫 𝐵))
7	1, 4, 6	e01 39233	. 2 ⊢ (   𝐴 ∈ 𝐵   ▶   {𝐴} ∈ 𝒫 𝐵   )
8	7	in1 39104	1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2030  Vcvv 3231   ⊆ wss 3607  𝒫 cpw 4191  {csn 4210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-pw 4193  df-sn 4211  df-pr 4213  df-vd1 39103

This theorem is referenced by: (None)