Theorem snelpwrVD 45567

Description: Virtual deduction proof of snelpwi 5424. (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 5409	. . 3 ⊢ {𝐴} ∈ V
2		idn1 45311	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
3		snssi 4750	. . . 4 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)
4	2, 3	e1a 45364	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   {𝐴} ⊆ 𝐵   )
5		elpwg 4564	. . . 4 ⊢ ({𝐴} ∈ V → ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵))
6	5	biimprd 251	. . 3 ⊢ ({𝐴} ∈ V → ({𝐴} ⊆ 𝐵 → {𝐴} ∈ 𝒫 𝐵))
7	1, 4, 6	e01 45428	. 2 ⊢ (   𝐴 ∈ 𝐵   ▶   {𝐴} ∈ 𝒫 𝐵   )
8	7	in1 45308	1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142  Vcvv 3454   ⊆ wss 3904  𝒫 cpw 4561  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-un 3909  df-ss 3921  df-pw 4563  df-sn 4589  df-pr 4591  df-vd1 45307

This theorem is used by: (None)